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1 

2 

3 

4 

5 

6 

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A   CLASS    IN    GEOMETRY: 


LESSONS   IN  OBSERVATION 
AND   EXPERIMENT. 


BY       ^ 

GEORGE  ILES'. 


NEW  YORK  AND  CHICAGO; 

E.  L.  KELLOGG  &  CO, 


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Copyright,  1894,  BT 

GEORGE  ILES. 


A  CLASS  IN  GEOMETRY. 


The  Study  of  Geometry  can  begin  with  Observation. 

—-The  general  disfavor  witli  which  my  schoolfellows  used 
to  open  Euclid  is  ii  vivid  remembriuico  of  my  boyliood. 
In  vain  did  the  teacher  say  that  geometry  underlies  not 
only  architecture  and  engineering,  but  navigation  and 
astronomy.  As  we  were  never  given  any  illustration  of 
this  alleged  underlyiiig  to  make  the  fact  stick  in  our 
minds,  but  were  kept  strictly  to  theorem  and  problem, 
Euclid  remained  for  most  of  us  the  driest  and  dreariest 
lesson  of  the  week.  Not  so  with  me;  for  geometry  was 
my  favorite  study,  and  the  easy  triumph  of  leading  the 
class  in  it  was  mii.e.  As  years  of  business  I'fe  succeeded 
my  school  days  I  could  not  help  observing  a  good  many 
examples  of  the  principles  set  forth  in  the  lines  and  figures 
I  had  conned  as  a  boy :  examples  which,  had  they  been 
presented  to  my  schoolfellows,  would  ;  iaiuly  have  some- 
what diminished  Euclid's  unpopulai:  .  In  fulness  of 
time  it  fell  to  my  lot  to  be  concerned  in  die  instruction  of 
three  boys,  one  fourteen  years  of  age,  the  second  twelve 
years  of  age,  the  third  a  tew  months  younger.  In  think- 
ing how  1  might  make  attractive  to  them  what  had  once 
been  my  best-enjoyed  lessons,  I  took  np  my  ink-stained 
Euclid,  Playfair's  edition.  A  glance  at  its  pages  dispos- 
sessed me  of  all  notion  of  going  systematically  through 
the  propositions, — which  took  on  at  that  moment  a  partic- 

3 


i*i|jfttwi>'"  jyj^^"'!" 


^^KSSi^iStHB 


.  A  CLASS  IN   CF.OMr.TRY. 

ularlv  riKi.l  look,  as  if  their  connection  with  the  worl.l  of 
fact  .u  I  life  w.«  of  the  ron.otcHt.     Why.  I  lhou«ht.  no 
akeriint  fron.  the  .no.hM-n  way  of  Ht.nlying  i.hyHU'H  and 
chemt  ry?     If  a  hoy  g^'tH  a  hotter   idea  of  an  electro- 
«    fron.  a  real  electru-n.agnet  than  fron.  a  p.cture.  or 
:r  dlly  understand,  the  chief  ^Y!^^^^^^^^^ 
when  he  sees  wood  ami  ir..n  hurned  in  it  than  w htn  I.e 
ivh  ars  orreadB  ahout  it«  con.bustivo  energy,  why  not 
:W  geometry  embodied  in  a  fact  before  Btatn.g  .t  .n 

i^rlict  pincipio ?  '>-<^'''"/"»;-n''ar;;'tt,n:d 

in  putting  book  and  bhickbuard  last  instead  of  tnst,  I  made 

'^ 'rtise-lot  and  Two  Fields  tell  us  Much.-Takin, 

thf  hoys  for  a  walk,  1  drew  their  attention  to  the  shape  of 

i:t'on  whieh  their  house  stood.     ^^^^^^^^^ 

thrice  its  width,  and  it  was  surrounded  I'Y  '\l«*/j"''- 

remX  home  I  asked  the  boys  to  make  drawings  of  the 
wTfieds  showing  to  a  scale  how  they  dilTered  in  size 

mg  1  This  talk  accomplished,  they  drew  a  diagram  of 
h  ';ouse.lot,andof  asquare  equal  in  area  to  the house-lo 

W  thafoot-r'ule  passed  around  the  latter  ^-g^m    t - 

Boon  clear  to  them  that  if  ^^^^  ^^^yf'' ""["^'^^lllZ 
eaual  some  fencing  would  be  saved.     [Fig.  2.]     Ihe  nex 
Zstion-ls  whether  any  other  form  of  lot  having  stra.g  t 
E  could  be  enclosed  with  as  little  fence  as  a  square 
Re    angles    and    triangles  were    drawn    in    considerable 
■  faltV  aul  their  areas  computed,  only  to  confirm  a  bus- 
*      ;     on  Z  boys  had  entertained  from  the  first,--that  of 
[o  to   pvactic  ble  form  square  ones  need  least  fencing.    In 


-■■>;  WBBlrtlwrw  J 


A  ri.ASS   IN  r.P.OMETRY. 


with  tlio  world  of 
iiy,  I  llioii^ht,  not 
(lying  pliyHics  uml 
leu  of  un   oloctro- 
from  a  picture,  or 
icloriHtic  of  oxygen 
II  it  tluiu  when  ho 
ire  energy,  why  not 
before  stilling  it  in 
:hiit  couUl  be  done 
tend  of  tirst,  I  made 

us  Much — Taking 
ion  to  the  shape  of 
Is  depth  was  nearly 
ed  by  a  low  fence, 
srn  snburb  of  Mon- 
•nced  lots  and  fields. 

number,  we  walked 
blished  the  fact  that 

area  of  one  was  an 
ten  acres.  When  we 
lake  drawings  of  the 
hey  dilTered  in  size. 
ey  drew  a  diagram  of 
area  to  the  house-lot. 
latter  diagram  it  was 
sides  of  the  lot  were 

[Fig.  2.]  The  next 
of  lot  having  straight 
le  fence  as  a  square, 
iiwn  in  considerable 
nly  to  confirm  a  sus- 
om  the  first,— that  of 
leed  least  fencing.    In 


comparing  tlicir  notes  of  the  jiunil)or  of  paces  taken  in 
walking  around  the  two  sfpiare  ilelds,  a  fact  of  some  im- 
portance  came   out.      While  the   larger   field    containeil 


Ifj.l. 


nearly  seven  times  as  mnch  land  as  the  smaller,  it  needed 
only  about  two  and  a  half  times  as  much  fence  to  surround 
it.    Taking  a  drawing  of  the  larger  enclosure,  I  divided 


F..4.«.. 

it  into  four  omial  parts  by  two  lines  drawn  at  right  angles 
to  each  other.  It  needed  only  a  moment  for  the  boys  to 
perceive  that  these  lines  of  division,  representing  as  they 


f 


5  A  (I  A5^!^  IN  r.F.OMF.TRY. 

,li,l  HO  mnd.  now  boumlary,  explai,...!  vvl.y  tl...  h.uuU  floM 
I  nvoporliunutoly  to  u.va  s..  uun-h  Ion,..- a  fe.u-e  Ih  u 

rl  m  duly  oucloH..!.  it  wu«  n.y  to  ..o  how  a  funnor 
whob  "1><  tl.' wl,ol..  ,.n.nk.r.  wore  ho  to  con.l.mo  l.c.n 
•  ,  l-t.-Lof  huul..ouUl  dinnml  hovou  oighthH  of  ,ho 
f",  g.  l»uring  a  journoy  hy  stoanu.r  fro.n  M..ntroal  to 
U  I  .linH-l'l  the  l.oy«'  attention  to  the  ..sadvuntng  j 
OS  ^vay  in  which  nnmy  of  the  farnm  had  been  d.vulod 
•I  "'l.  h>ng  and  narrow.  «  T-t  ^ike  a  row  of  chess 
aonares  run  together,"  Baid  one  of  the  hidH. 

Trom  Fact  to  Law.-When  a  good  nmny  examples  had 
i,,,,XHl  our  tlrst  lenson  on  their  nnnd^  pretty  thoroughly 
tl     boys  draw  on  large  sheets  of  paper  two  squares 
es  ec    v^ly     i.K^l>  and  1  foot  in  length      Heneath  these 
Zre     hoy  wrote:  "  I  inch  Bc,uare  has  4  l.neal  .nches  for 
o  nda,T      f..ot  square  has  48  lineal  inches  for  boundary ; 
Un  h   quare  has  1  square  inch  for  area,  1  foot  square  has 
44    quire  inches  for  area.     Plane  tlgnres  of  the  same 
om  have  boundaries  varying  direclly  as  the.r  like  hnear 
r  nslns  (length  and  breadth) ;  they  have  areas  varying 
as  the  m^re  «?  their  like  linear  dimensions       It  proved 
rowver,  that  while   the  boys  knew  this  to  be  true  of 
souares    they  could  not  at  first  comprehend  that  it  , 
eS  y  true  of  other  plane  figures.    They  drew  equ.latera 
dtie;-  triangles,  and  in  measuring  their  -^es  -d  area 
*■  -,,0,1  tlmt  thev  conformed  to  the  rule;  but  1  was 
rr       lack  al  tt le  when  the  oldest  boy  said,  "It  isn't  so 
'ircir  les  is  i^^  "        iB  Joubt  was  duly  removed,  but  the 
::ma  rs  W  ^^^^^^^  easy  it  is  to  make  words  outrun  idea^- 
how  bard  it  is  for  a  young  mind  to  recognize  new  cases  of 
a  Ineral  law  which   n  other  examples  is  quite  familiar 

Ses  Terrestrial  and  Celestial- One  chilly  evening 
the  room  in  which  my  pupils  and  I  sat  was  warmed  by  a 


m 


.t..--*vi,-    -  ■,. . ii-.>:MisSiSt|«Wi  ■ "» ..+' 


,•  tlu<  siiuiU  field 
;«'!•  II  teiico  llmii 
I'l-vi'd  as  another 
|iiiire8  ri'proHciit- 
I'O  how  a  fui-Jiicr 
o  c'oiiihiiio  tla-m 
u  eightliH  of  tho 
'om  Moiitroiil  to 
he  (lisadvantnj^e- 
lad  heciJ  divided 
e  a  row  of  chess 

i. 

ny  examples  had 
n-etty  thoroughly, 
lapor  two  squares, 
.     Heneath  these 
[  lineal  inches  for 
lies  for  houndary; 
1  foot  square  has 
II res  of  the  same 
3  their  like  linear 
lavc  areas  varying 
ions,"     It  proved, 
lis  to  be  true  of 
rehend   that  it  is 
•y  drew  equilateral 
eir  sides  and  areas 
3  rule;  but  I  was 
y  said,  "  It  isn't  so 
y  removed,  but  the 
rds  outrun  ideas, — 
ignize  new  cases  of 
s  quite  familiar. 
One  chilly  evening 
,t  waa  warmed  by  a 


A  CLASS  IN   GKOMhTRY.  7 

prate-fire.  Shaking  out  womo  small  live  ooal-s,  i  bade  tlio 
boys  observe  wliicii  of  them  tunitul  black  soonest,  'i'hey 
were  quick  to  seo  that  tho  smallest  did,  but  they  were 
unable  to  tell  why,  until  F  broke  a  large  glowing  coal  into 
a  Hcoro  of  fragments,  which  ahnoHt  at  once  bi-came  black. 
[Fig.  3.J     Then  one  of  l!iem  cried,  "  Why,  swashing  that 


coal  gave  it  more  surface  ! "  This  young  scholar  waa 
studying  the  elements  of  astronomy  at  school,  so  I  had 
him  give  us  some  account  of  how  llif"  planets  diifer  from 
one  another  in  size,  how  the  moon  compares  with  tho  earth 
in  volume,  and  how  vastly  larger  than  any  of  its  worlds  is 
the  sun.  Explaining  to  him  the  theory  of  tho  solar  sys- 
tem's fiery  origin,  I  shall  not  soon  forget  his  keen  delight 
— in  which  the  others  presently  shared — when  it  burst 
upon  him  that  because  tho  moon  is  much  smaller  than  the 
earth  it  must  be  much  cooler;  that,  indeed,  it  is  like  a 
small  cinder  compared  with  a  large  one.  It  was  easy  to 
advance  from  this  to  understanding  why  Jupiter,  with 


8  A  CLASS   IN   GEOMETRY. 

eleven  times  the  diameter  of  the  earth,  still  gjo^  f^^^^Jy 
In  Ihe  sky  by  its  own  light,  and  tb.n  to  note  that  he  sun 
pou  s  S  ils  wealth  of  heat  and  light  be-ause  the  im- 
mensity of  its  bulk  means  a  comparatively  small  surface  to 
Td  ■  te  from.  Here,  incidentally,  we  can.e  upon  now 
"ro  d  fo  satisfaction  with  the  small  world  we  n.hab.t: 
only  he  surface  of  planets  can  sustain  life,  so  that  ni  pro- 
p  rlion  to  its  volume  the  earth  bears  eleven  t.mes  the 
plaTund  animals  it  would  were  it  swollen  to  the  gu-th  of 

'tbLl  Blocks  make  the  Law  Plain  and  Gear  Jo 
make  the  law  concerned  in  these  examples  ^efin  te  a.^ 
clear,  I  took  eight  blocks,  each  an  inch  -b«,jd  ».ad  t^ 
bovs  tell  me  how  much  surface  each  had-6  square 
Ses  Building  the  eight  blocks  into  one  cube  they 
then  counted  the  square  inches  of  its  ^yf-^-^^^;^/;; 
times  as  many  as  those  of  each  separate  cube,.    With  twenty 

even  blocks\uilt  into  a  cube,  that  structure  was  found 
to  have  a  surface  of  54  square  inches-nme  times  that 

of   ach  component  block.    [Fig.  4.]    As  the  blocks  under- 


went the  building  process,  a  portion  of  their  surfaces  came 
Tto  contact  ai/thus  hidden  could  not  count  m  the  outer 
into  confaci,  a  observation  and  comparison 

surfaces  of  the  large  cuoes.    v  exactly  what 

brought  the  boys  to  the  rule  which  told  «^,f^^^y J^ 
.rop'rtion   of   surface  remained  exposed  J^^j  w  o  e 
"Solids  of  the  same  form  vary  in  surface  as  the  squau, 
Jd  fu  ctt.uts  as  the  cube,  of  their  like  dimens.ou.. 


e  i»=-r^jte>  I,  r*lW«  *.>i«^»*^^'*4'4.-«t 


-r-w— *--»»^ 


till  glows  faintly 
lote  tliat  the  sun 
bcciiuso  the  ini- 
;  small  surface  to 
oanie  upon  now 
rorld  we  inhabit: 
e,  so  that  in  pro- 
eleven  times  the 
en  to  the  girth  of 

I  and  Clear — To 

iples  definite  and 
3ube,  and  had  the 
h  had— 6  square 
;o  one  cube,  they 
surface— 24:  four 
be..  With  twenty- 
ructure  was  found 
, — nine  times  that 
3  the  blocks  nnder- 


their  surfaces  came 
t  count  in  the  outer 
ion  and  comparison 
told  exactly  what 
ased.  They  wrote, 
•face  as  the  square, 
ir  like  dimensioue/' 


A  CLASS  IN  GEOMETRY.  9 

They  were  glad  to  note  that  the  first  half  of  their  new  rule 
was  nothing  but  their  old  one  of  the  farms  and  fields  over 
again. 

Gravity  at  the  Surfaces  of  the  Sun  and  the  Earth. — 
Keturning  next  day  to  the  subject  of  astronomy,  I  took  a 
terrestrial  globe,  and  rotating  it  in  a  variety  of  i)lanes  made 
it  clear  that  the  centre  of  the  sphere  was  the  only  point  at 
rest  in  the  mass,  and  that  therefore  the  whole  volume  of  a 
planet  or  a  star  may,  in  astronomical  study,  be  considered  as 
concentrated  at  the  centre.  The  eldest  boy  was  then  asked 
to  refer  to  his  text-book  and  inform  us  what  the  radius  of 
the  sun  is  compared  with  that  of  tlie  earth.  He  answered, 
about  lOJ)  times  as  much.  lie  also  found,  on  farther  refer- 
ence, that  the  specific  gravity  of  the  sun  is  about  one  fourtli 
that  of  the  earth.  Taking  the  first  figure,  without  frac- 
tions, he  computed  the  volume  of  the  sun  in  units  of  that 
of  the  earth  as  1,295,029.  He  knew  tluit  gravity  dimin- 
ishes in  intensity  as  tlie  square  of  the  distance  through 
which  it  is  exerted.  I  now  inquired  how  much  a  mass 
weljThir:g  one  pound  on  the  surface  of  the  earth  would 
weigh  if  taken  to  the  surface  of  the  sun.  Dividing  1,295,- 
029  by  11,881  (the  square  of  109),  and  by  4  (the  specific 
gravity  of  the  earth  as  compared  with  that  of  the  sun),  ho 
rendered  27^  as  the  answer.  With  a  little  patience  not 
only  the  eldest  boy,  but  the  other  two  boys,  saw  the  analogy 
between  this  case  and  that  of  the  large  cube  bnilt  out  of 
the  twenty-seven  little  ones.  For  a  reason  which  would 
become  clear  in  a  future  lesson,  I  asked  them  to  ponder 
the  principle  that  the  very  fact  of  bigness  implies  that  the 
surface  of  an  attracting  mass  is  far  from  its  centre,  and 
that  despite  the  immense  rate  at  which  the  volume  of  a 
sphere  increases  as  its  diameter  lengthens,  a  sphere's 
power  of  attraction  at  its  surface  increases  only  in  thu 
direct  ratio  of  the  diameter's  increase  of  length, — as  we 
bad  seen  in  ascertaining  the  comparatively  small  augmeu- 


.  vs-,i^i3sisifeBr»<i-r*- 


,0  A  CLASS  IN   GEOMETRY. 

tation  of  gravity  whioh  would  be  observable  in  a  pomul 
taken  from  the  earth  to  an  orb  much  more  than  a  million 

times  as  big.  _         ^  «i.^««.*- 

A  Vial  Illustrates  the  Path  of  an  Ocean  Steamer.- 

Aa  the  law  at  which  we  had  now  arrived  is  one  of  the 
most  important  in  geometry,  I  took  pains  to  illustrate  it 
in  a  variety  of  ways.     Taking  a  long,  narrow  vial  of  dear 
glass,  nearly  filled  with  glycerine,  and  corked,  I  passed  it 
round,  requesting  each  of  the  boys  to  shake  it  smartly, 
hold  it  upright,  and  observe  which  of  the  bubbles  came  to 
the  surface  first.    [Fig.  5.]    All  three  declared  that  the 
biggest  did,  but  it  was  a  little  while  before 
H-^-       they  could  be  made  to  discover  why.     Ihey 
had  to  be  reminded  of  the  cinders  and  the 
building-blocks  before  they  saw  that  the  com- 
paratively large  surface  of  a  small  bubble  re- 
tarded  its  motion   through  the  liquid.     Ihe 
next  day  we  visited  the  wharves  of  Montreal, 
and  pacing  alongside  several  vessels,  jotted 
down  their  length.     In  response  to  questions, 
the  boys  showed  their  mastery  of  the  principle 
which  decides  that  the  larger  a  ship  the  less 
in  proportion  to  tonnage  is  the  surface  at 
which  the  vessel's  motion  is  resisted,     boing 
aboard  an  Allan  liner,  of  five  thousand  tons 
burden,  we  descended  to  the  engine-room.   We 
next  vlbited  a  steamer  of  somewhat  less  than 
one  thousand  tons  and  inspected  her  engines,- 
engines  having  proportionately  to  PO^«r  much 
larger  outer  surfaces  whereat  to  lose  heat  than 
those  we  had  seen  a  few  minutes  before  in  the 
Allan    steamship.     When  their  experiments 

with  the  vial  were  recalled,  the  boys  were 

t^leasantlY  surprised  to  find  that  the  largest  bubble  and 
rieo    an  racer  come  first  to  their  respective  ports  by 


I II  mrmwiiiiii    iiiiirriiiiiii  ' 


ty^-M^J^'-^H'T 


A   CLASS    IN   GEOMETRY. 


II 


ible  in  a  pound 
re  than  a  million 

tean  Steamer.— 

d  is  one  of  the 
aa  to  illustrate  it 
•row  vial  of  clear 
irked,  I  passed  it 
hake  it  smartly, 
!  bubbles  came  to 
leclared  that  the 
ttle  while  before 
3ver  why.     They 
cinders  and  the 
saw  that  the  corn- 
small  bubble  re- 
the  liquid.    The 
rves  of  Montreal, 
■al  vessels,  jotted 
)nse  to  questions, 
ry  of  the  principle 
er  a  ship  the  less 
is  the  surface  at 
5  resisted.     Going 
ive  thousand  tons 
engine-room.   We 
)mewhat  less  than 
3ted  her  engines, — 
;ely  to  power  much 
,t  to  lose  heat  than 
ttutes  before  in  the 
their  experiments 
3d,  the  boys  were 
largest  bubble  and 
respective  ports  by 


asis»««'  te«4i*te8SifiS3S 


virtue  of  their  identical  quality  of  bigness,  by  reason  of 
the  economies  wliich  dwell  with  size. 

The  Biggest  Engines  are  Best.— Ikit  the  ocean  racer 
had  an  advantage  in  the  })ignc.ss  of  its  engines  which  was 
not  to  be  read  in  a  simple  ghmco  at  their  exteriors.     The 
boys   know    that   iu    striking  an    anvil   botii   anvil   and 
hammer  become  iioated,  for  at  a  hliiLksmitii's  sliop  near  by 
thoy  had  had  evidenoo  of  tlie  fact;  and  in  descending  a 
slope  of  Mount  Uoyal  an  anipty  cart   had  one  day  l)i'i'n 
observed  checked  in  speed  by  a  brake;  a  toucli  on  tliis 
brake  when  the  cart  was  stopped  had  informed  them  again 
tliat  mechanical  work  can  be  converted  into  heat.     Before 
we  left  tlie  steamer  I  told  the  boys  that  if  we  had  a  ther- 
mometer connected  with  the  cylinder  of  the  engine  wo 
sliould  see  that  the  steam  in  expanding,  in  doing  work, 
would  fall  iu  temperature:  it  stood  to  reason  tlnit  if  the 
power  of  the  engine  were  applied,  not  in  moving  tlie  ship 
but  iu  lifting  trip-hammers,  the  heat  whicli  these  ham- 
mers would  create  in  falling  on  anvils  would  be  lost  by  the 
working  steam.     Only  the  eldest  boy  could  follow  me  in 
this  statement;  it  was  some  weeks  before  the  others  could 
do  so  ;  but  all  three  were  gradually  brought  to  understand 
the  great  gain  which  comes  of  making  a  steam-cylinder  as 
large  as  possible.     I  said  that  its  contents  in  expanding  be- 
come cooled,  the  steam  thus  cooled  robs  the  cylinder  itself 
of  some  of  its  heat,  so  that  when  the  next  working  charge 
of  steam  comes  in,  that  charge  is  chilled  by  the  walls  of  the 
,  cylinder  and  loses  part  of  its  working  power,— since  work- 
ing power  depends  on  high  temperature.     Hence  the  itn- 
portance  of  using  the  strongest  material,  as  steel  instead 
of  iron,  so  as  to  make  engines  and  cylinders  of  the  utmost 
size,  reducing,  comparatively,  the  surfaces  at  which  in- 
jurious cooling  takes  place.     Before  we  left  the  steamer  I 
added  a  word  on  the  necessity  for  compactness  iu  engines 
and  machinery  erected  \Yhere  space  ia  scant,  as  aboard 


la 


A  CLASS   IN   GEOMETRY. 


\ 

5 


ship,  and  told  the  boys  that  tlioy  would  find  the  best  en- 
gines for  work  on  laud  modelled  on  the  compact  form  of 
en  chines  built  for  service  at  sea.  (At  the  World's  Colum- 
bia'n  Exposition  an  AUis  steam-engine  of  4000  horse-power 
was  shown.  A  firm  in  England  is  now,  1894,  building  a 
gas-engine  of  (iJO  iiorse-power,  in  dimensions  hitherto  un- 
tiltempted.  In  the  substitution  of  steel  for  iron  an  im- 
mense economy  in  weight  is  enjoyed  by  the  builder  not 
only  of  engines,  but  of  machinery,  buildings,  and  bridges. 
A  similar,  tbough  less  important,  advance  is  promised  in 
the  current  experiments  with  nickel  and  other  alloys  of 
steel  and  of  aluminium.  These  alloys  may  yet  give  the 
world  higher  towers  and  longer  bridge-spans,  thinner  and 
therefore  more  efiicient  boilers,  and  may,  while  providing 
larger  engines  and  machines,  render  them  capable  of  higher 
speeds  than  those  at  present  safe.) 

Balloons  Small  and  Great  — One  afternoon  the  boys 
and  I  while  walking  together  espied  a  street- vendor  with 
a  supply  of  gaudy  toy  balloons.  A  balloon  hought  for  the 
youngest  of  our  party,  I  dare  say  that  the  little  fellow 
wiis  pretty  confident  that  there  was  no  Euclid  in  that 
plaything.  It  proved  otherwise.  That  evening  he  calcu- 
lated how  much  the  lifting  power  of  the  balloon  would 
gain  on  its  surface  were  its  dimensions  multiplied  one 
thousand  or  ten  thousand  fold,— step  by  step  reaching  the 
conclusion  that  if  air-ships  of  that  kind  are  ever  to  be 
manageable  in  the  face  of  adverse  winds,  they  must  be 
made  vastly  larger  than  any  balloons  as  yet  put  together. 

A  Store  is  Visited.— Not  far  from  home  stood  a  largo 
store,  displaying  a  miscellaneous  stock  of  groceries,  fruits, 
dry-goods,  shoes,  and  so  on.  As  we  cast  our  eyes  about  its 
shelves,  counters,  and  floor,  we  saw  many  kinds  of  pack- 
ages—cans of  fish,  marmalade,  and  oil;  glass  jars  of  pre- 
serves and  olives;  boxes  of  rice  and  starch;  large  paper 
sacks  of  flour.    Outside  the  door  stood  half  a  dozen  empty 


A   CLASS   IN   Gi:OMKTRY. 


»3 


find  the  best  en- 
loinpuct  form  of 

World's  Colum- 
1000  horse-i)ower 

1894,  building  a 
ions  hitherto  uu- 

for  iron  an  ini- 

the  builder  not 
iigs,  and  bridges. 
;e  is  promised  in 
3  other  alloys  of 
nay  yet  give  the 
)ans,  thinner  iind 
,  while  providing 
capable  of  higher 

ternoon  the  boys 
treet-vendor  with 
»n  bought  for  the 

the  little  fellow 
o  Euclid  in  that 
evening  he  calcu- 
he  balloon  would 
18  multiplied  one 
step  reaching  the 
id  are  ever  to  be 
ids,  they  must  be 
yet  put  together, 
ume  stood  a  largo 
»f  groceries,  fruits, 

our  eyes  about  its 
iiy  kinds  of  pack- 

glass  jars  of  pre- 
;arch;  large  paper 
lialf  a  dozen  empty 


barrels  and  packing-cases.  It  certainly  seemed  as  if  the 
ite  IS  o?  paper,  glass,  tin,  and  lumber  for  packages  must 
J;  e  1  u'^ely  in  o  the  cost  of  retailing.  O.ie  after  another 
lb  y  "discovered  that  the  store  was  giving  them  their 
^;::Linanewform:theysawthatthel.^ora^ror 

box  the  less  material,  in  proportion  to  capamty,  .t  noakd 
f  1  its  manufacture.     The  contents,  too,  o     he  ja,.  sa  ks 
and  boxes  repeated  the  familiar  story.    The   coffee   had 
in  ground  from  the  bean  that  it  n.ight  be  the  mo. 
effectively  boiled;  and  certain  brands  of  white  sugnr  had 
;!::   ::Werized  from  the  lump  tl-t  they  ,n.ght     e    ho 
more  quickly  dissolved  at  the  breakfast-  able      On  the 
Zh      0  the  bovs  were  led  to  discern  that  form  as  we 
r'       s  an  element  in  economy.     J«st  as  larms  square 
Inne    -eed  least  fence,  so  a  cubical  package  needs  least 
^Xill  io  make  it,  and  tins  of  cyli;>^Hca|  ^orm  ^quin^ 
least  metal  when  of  equal  height  and  breadth.      i^^-f^ 
this  of  some  importance  when  we  learn  j-^  du.n^lS^^ 

aan  <wi()  ()(«)  cans  were  made  m  the  United  states.) 
'TJ^Sern^^J^^^ment  in  his  Use  of  Geometry. 
_The  boys  well  knew  that  of  plane  figures  the  circle,  he 
Bectio;  at  right  angles  of  a  cyUude.^  .  the  one  1^^^^^^^^ 
1p.mt  boundary  in  proportion  to  surface;  they  tlcielore 
f  fnnd  vthv  cvlimlrical  pipes  are  used  for  water  and 
::' "ts  we  waM  Ing  we'noticed  here  and  there  cylin 
gas.    AS  we  wa  J'         ^  ^^  ^f  houses  with  the 

drical  --l-f;XX  boys  observe  whether  these  con- 


^sgr- 


M 


A  CLASS  IN  GEOMETRY. 


question  of  judicious  form  was  ono  dopcndiu!;  upon  cir- 
cunistiincos  to  bo  ciirofully  considered  in  eueli  case.  Of 
tliis  wo  had  another  illuntration  when  ono  morning  wo 
came  upon  a  hirge  ice-house  near  tlie  river.  It  was  very 
unlike  a  cube  in  its  outlines,  although  a  cube  is  the  form 
bounded  by  planes  which  has  least  surface  in  comparison 
with  capacity.  The  ice-house  had  a  frontage  twice  the 
extent  of  its  depth;  its  height  was  scarcely  half  its  depth; 
its  roof  was  not  Hat,  but  slotting.  Uy  a  series  of  (piestions 
as  to  the  strength  of  walls  and  the  cost  of  building  them 
BO  thi(!k  that  tliey  can  be  high;  as  to  the  exiteiise  of  lift- 
ing and  lowering  ice  in  a  lofty  building;  the  necessity  for 
making  a  roof  aslant  if  it  is  to  shed  rain  and  melted  snow, 
— we  camo  to  some  of  the  reasons  why  the  builder  had  de- 


'tili 


I 


parted  from  the  cubical  form  a\  this  structure.     But  why 
was  the  ground  plan  oblong  instead  of  square?     liecau^^L' 
the  lot  was  of  that  configuration,  and  because  that  par- 
ticular lot  had  advantages  of  locality,— it  lay  between  the 
ice-field  and  the  city,  and  was  so  near  the  city  that  tlu' 
cost  of  cartage  was  minimized.     About  a  mile  from  the 
ice-house  stood  a  large  sewing-machine  factory.     It  also 
did  not  conform  to  a  cubical  outline  :  another  series  of 
questions  brought  out   the   desirability   of  its   extended 
frontage  for  light,  and  of  building  a  structure  like  this 
one,  occupying  costly  land,  of  more  stories  than  if  it  were 
built  on  cheap  land.     These  examples  were  two  of  several 
Avhich  I  gave  the  boys  to  demonstrate  that  it  is  a  mistake 
to  over-simplify  a  jji-oljlem,  and  that  judgm^wt  must  direct 


'«WtaHtgW»i»IH«g^'WB!Ba^^ 


A  CLASS  IN   GfiOMETRY. 


15 


lendiii!,'  upon  eir- 
11  oiuih  case.     Uf 

0110  inoi'uiiig  wo 
iver.  It  was  very 
I  cube  is  tlie  form 
ice  in  comparison 
•outage  twice  tlio 
']y  lialf  its  deptli; 
series  of  (picstions 

of  hiiildiiig  them 
10  expense  of  11  ft- 
;  tiio  necessity  for 

and  melted  snow, 
lie  builder  Jiud  do- 


ructure.     But  wliv 


square 


?     Because 


because  tliat  par- 
-it  lay  between  tlie 
'  the  city  that  the 
t  a  mile  from  the 
18  factory.  It  also 
:  another  series  of 
y  of  its  extended 
structure  like  this 
iries  than  if  it  were 
were  two  of  several 
that  it  is  a  mistake 
ilgmiiiit  must  direct 


calculation  and  pass  upon  its  results  if  a  w.so  ;lec'«'»»  ^^ 
to  be  reached.  As  they  grew  older  they  found  th>,^  the 
conunon  case  of  disagree.nent  between  practice  md  theory 
is  due  to  theory  not  taking  account  of  all  he  facts  that 
>vhenever  a  theory  does  take  account  of  all  the  facts,  any 
practice  at  variance  with  that  theory  is  wrong,  and  should 

be  abandoned.  .  ,, 

Victoria  Bridge. -Our  next  lesson,  m  somewhat  the 
same  line  as  the  last,  was  one  for  lack  of  which  not  a  few 
inventors  and  designers  have   wasted   time   and   nioney 
Taking  the  trio  to  Victoria  Bridge,  we  inquired  of  the 
custodfan  the  length  of  its  central  span.        .s  reply  was 
35->  feet      When  I  asked  the  boys  how  matters  would  be 
changed  if  the  span  were  twice  as  large  in  all  three  dimen- 
sions'they  soon  perceived  that,  while  increased  in  strength 
by  increale  of  bl-eadth  and  thickness,  it  would  be  heavier 
by  added  length  as  well.     On  our  return  we  compared  two 
small  beams  ditfering  in  each  of  their  three  d-ensmns  as 
1  and  2.  serving  to  make  manifest  why  it  often    lappen 
that  a  design  for  a  bridge,  roof,  or  machine,  admm^^^^^^^ 
in  a  small  model,  fails  in  the  large  dimensions  of  practical 
onstruction,  and  may  even  fall  to  pieces  by  its  own  weight. 
FFig  7  1      For  weight  increases  as  the  cube,  and  strength 
only  as  the  cross-section,  or  square,  of  like  dimensions. 


Why  the  Sand-blast  is  Efficient—One  day  we  went  to 
a  factory  where  designs  were  being  marked  upon  glass  by 
a  and  bJt.  The  apparatus  was  of  the  simplest:  from  a 
hopper  near  the  ceiling  sand  fell  in  a  narrow  Btream 


"WTH 


IJIUKIJIJIIIIIIIII 


■<niv 


'mm'^ 


16 


A  CLASS  IN  GEOMF.TRY. 


upon  tlio  panes  or  goblota  exposed  to  its  action.      IIiul 
Bundstonc    in    lun)]>s,   .ssiy  us   lur^o   as   |)ltiyin^    marbles, 


y.^.e. 


been  dro})i)ed  ii])on  tlie  glass  there 
would  only  huvo  been  harmful  frae- 
ture.  As  it  was,  each  particle  of 
sand  weighed  too  little  in  projjor- 
tion  to  its  striking  surface  to  do 
any  more  than  the  desired  work  of 
detaching  a  tiny  chip  from  the  face 
of  the  glass  it  fell  upon.  [BMg.  8.] 
Common  Dust  and  Flour  Dust.— 
Not  long  afterward  a  roofer  was 
called  in  to  make  needful  repairs 
at  the  house  where  two  of  the  boys 
lived.  We  went  with  him  to  the 
roof,  and  found  the  gutter  choked 
with  nuid.  How  had  it  got  there  ? 
A  glance  at  the  roof — an  iron  one — 
showed  it  covered  with  dust  which 
the  next  shower  would  sweep  into 
the  gutter.  Dust  particles  are 
extremely  small  and  fine — much 
smaller  and  finer  than  sand;  and 
did  not  this  explain  how  the  wind 
had  been  able  to  take  hold  of  them 
and  carry  them  far  up  into  the 
air?  Here  I  said  a  Avord  about  the  terrific  explosions  in 
the  flour-mills  of  Minneapolis,  due  to  nothing  else  than  to 
flour  reduced  to  a  dust  so  fine  as  readily  to  form  with 
air  an  explosive  mixture.  Disastrous  explosions  in  the 
coal-mines  of  Austria  liave  on  investigation  been  proved 
chargeable  to  coal-dust  as  fine  as  powder,  which  has  found 
its  way  through  the  air-shafts.  When  the  boys  afterward 
took  up  geology,  they  had  a  key  to  one  of  its  most  inter- 
esting chapters.    Dust  carried  by  winds  has  filled  lakes, 


"WW^ 


0  its  action.  Had 
s  playing  marbles, 
pon  tiio  glaM8  thero 
3  been  harmful  frac- 
is,  each  particle  of 
00  little  in  propor- 
king  surface  to  do 
the  desired  work  of 
f  chip  from  the  face 
ill  upon.  [EMg.  8.] 
t  and  Flour  Dust.— 
ward  a  roofer  was 
iko  needful  repairs 
lere  two  of  the  boys 
it  with  him  to  the 

the  gutter  choked 
vv  had  it  got  there  ? 
roof — an  iron  one — 
ed  with  dust  which 
r  would  sweep  into 
Dust  particles  are 
11  and  fine — much 
ler  than  sand;  and 
plain  how  the  wind 
0  take  hold  of  them 
11  far  up  into  the 
erriflc  explosions  in 
lothing  else  than  to 
iidily  to  form  with 
3  exi)losions  in  the 
gation  been  proved 
9r,  which  has  found 

the  boys  afterward 
3  of  its  most  inter- 
ids  has  filled  lakes, 


A  CLASS  IN   GhOMHTRY. 


'7 


covered  plains,  a.id  buried  cities  from  sight.  Thinning 
over  (Uk.ivcmI  rock,  nt roams  of  water  have  bonio  its  clay 
iKirliclcs  imich  farMu'r  tban  its  sand. 

Earth  as  Mud  is  easily  carried  by  Water.-In  early 
sprii..'  the  rills  and  streams  desceMding  tlie  slup"s  ol 
Mo.urt  U..val  are  visibly  charged  willi  nuul.  as  the  b..ys 
hud  repoati-dly  ol)serve.l.  I  told  them  that  ti>e  Mississippi 
is  not  a  clear  river  like  the  St.  {.awr.-nce,  an.l  gave  tliem 
some  of  the  surprising  ilgures  whieli  r.MM.nl  (be  .,"a"<'ty 
of  earth  in  a  state  of  <livision  as  line  mud  whieb  is  borne 
away  by  the  Father  of  Waters.  Near  its  mouth  tlu'  Missis- 
sippi sometimes  carries  in  suspension  as  much  as  tw..  pounds 

of  solid  matter  in  every  cubi.:  fo..t,-..ne  tl,irty.se.-o„d  of 
its  stream,-an  amount  s.,  considerable  that  it  en  ers  into 
the  computati.ms  of  the  geologist  when  he  studies  the  prin- 
cipal ways  in  which  the  water  of  the  globe  gains  upon  the 

^""En<.ineors  by  availing  themselves  of  this  carrying  action 
of  waiter  arc  able  to  deepen  watercourses  by  reducing  to 
fine  mud  the  surface  of  their  beds.  At  Tilbury  Dock,  in 
En-land,  this  process  has  been  adopted  as  on  y  one  fortieth 
I  Mostly  as  ordinary  dredging.  It  was  found  that  in  rapid 
eddies  water  could  lift  and  remove  clay,  stone,  and  gravel, 
of  much  higher  specific  gravity  than  itself,  when  these 
obstructions  were  in  small  enough  pieces. 

Reduction  to  Powder  often  Advantageous.-In  a  new 
and  improved  way  of  manufacturing  Portland  cement, 
Mr  Zome's,  the  ingredients  are  reduced  to  a  powder 
Ltead  of  being  brought  together  in  lumps  Ihe  supen- 
0  tTof  tlm  mfthod  and  its  saving  in  time  have  long  been 
h  nted  at  in  all  the  processes  of  grinding,  as  in  the  produc- 
tbu  of  ttour,-so  much  to  be  preferred  to  whole  or  even 
brkei  wheat.  And  be  it  remembered  that  in  reducing 
w^a  to  powder  the  miller  only  imitates  the  action  whoro 
Jy  his  own  teeth  break  his  mM  food  luLo  «mall,  digestible 


™C5lfey«i«i»»19AB«»','»'«MMISlli:''''"' 


|8 


A   CLASS   IN    C.I'.OMHTRY. 


,„„.t«.     Aswitl.  wlH-utso  with  tlu.f.rtili/,.Tsns,.l  im  (l,o 

vlu-ut-rM'M.      A  .rnlnry  n^n   houv-^    ...rirly   l-ruken   w.n" 
uxoawiththoourthbytla,  fanuH-;   tcMlnyal.urnl 

eitkiencv  uris.H  fmm  tl.o  ap,.Iintti.m  of  l,o,u.,s  „•..„,..  to 
,l„Kt.  An.l  tl.on.^h  very  f.ir  ITon.  l.'...-'  an  s.na  1  as  a 
,,i,,iclc"  of  <lnst.  it  is  partly  bmiuso  a  f,mMM  ut  wh.-at  m 
;,„,„  „,  ,i.H.  is  nmcl.  s.nall.T  tluu.  ..nlinary  ITnils  .,r  v.-.>- 
,nl,l..s-as  ai-l'l-s  <.r  I-taU-.s  tl.at  th.  .r>v;.ls  a,o  nun.  s 
,;;J,  foo<l.  'I'lu.ir  sn,all-...ss  ..al^i.s  tlu...  tu  .iry  .,.n.-kl 
,„„,    tl.on.n^l.ly,  so    that,    tln.y  aro   easily    p.vsiTNf.l    a.nl 

^'"HowRock  becomes  Soil.   -In  <-onnnon  will,  other  sin.ilar 

«i,.si,.no.al.cTnla.it,,,i.s,Monnt  UoyaliM  n>M,,ysn.n.so^ 

places  in  ca.ly  sprin,'  shows  how  In.st  can  -ieta-'h  ihe  ...  - 
ace  of  a  roek,  or  h.'eak  i,.to  r.-a:.n„onts  a  n.ass  ot  stone 
to  which  water  has  entei-e.!.     An.l  a.,  action  very  sun.lar 
is  exerted  by  the  force  ..f  growth  it«elf ;  we  couhl  ohservo 
in  several  phioes  a  stu.-.ly  sapling  cleaving  ap.irt  a  ,.ot  ...- 
conshlerahle  hit  of  rock,  wl.e.-o  a  seed  a  few  years  hefo  o 
had  taken  root  in  a  crevice.     Ml  this  we,,    to  shmv  the 
boys  how  hard  rock  gradually  bccon.es  so.l  by  v..tne  o 
foLs  wluch  hre.tk  up  large  n.asses  into  smaller  o,.es  a  d 
so  expose  i.ew  surfaces  to  dissolv.t.g  air  and  wat^ei.     Ihe  r 
atten  ion  was  furthermore  directed  to  the  -Ine  o     th 
plough,  the  harrow,  and  the  sharp-toothed  "  cult  nato, 
which  lift  and  stir  the  soil  of  a  farn,  or  gardcn.-creating 
fi-esh   surfaces  whereat   rootlets  can  feed,  and  rendenng 
easv  the  growth  of  these  thread-like  structures. 

Plants^nd  Insects  in  Farther  Illustration—  Aneighbor 
of  onrs  had  a  large  co,.servatory,  in  which.  ^^-^^^^^ 
exotics,  were  various  bristling  cacti.  \\  hen  t  e  boys 
Ltei  that  the  cactus  is  a  native  of  the  desert  they  s.^ 
at  once  its  adaptation  to  dry  air  and  a  harsh  so.l;  m  com- 
parison with  ordinn,-y  plants  its  surface  ,s  n.eagre,  and  is 
stem  very  fleshy.     ThU  while  according  to  Professor  Asa 


A   CLASS   IN   (1F.(>MI:TRY. 


»9 


izcrs  uscil  ill  tlio 
•cly  liniki'Ii  wiTO 
to-day  11  r«)iirf'>Iil 
lioiifH  liniiiiul  111 
\ti  as  small  as  a 
rniiii  (if  wheat  nr 
ary  I'm  lis  <»r  vcjif- 
ctifiil^  iii(!  man's 
III  Id  ilry  (iiiii'kly 
ily    pri'siM'Nfil    uiiil 

willi  Dtlicr  similar 
ill  iiiMiiy  scores  of 
'Uii  (It'tacli  lilt'  siir- 
;s  a  mass  of  stono 
action  very  similar 
;  wc  oould  ohservo 
iui;  apart  a  not  iti- 
a  few  yoiii''^  hoforo 
1  went  to  show  tlio 
IS  soil  by  virtue  of 

0  smaller  ones,  and 
•  and  water.  Their 
)  the  value  of  the 
othed  "  cultivator," 
r  jrarden,— creating 
eod,  unci  rendering 
■uctnres. 

ration. —  Aneighbor 

A'hich.  among  other 

.      When    the   boys 

the  desert,  they  saw 

1  harsh  soil ;  in  com- 
ice  is  meagre,  and  its 
mg  to  Professor  Asa 


Gray  an  ordinary  elm  in  full  leaf  has  live  acres  of  foliaj;e. 
One  afternoon  our  neighbor  showed  us  a  convolvulus  he 
had  just  received  from  Colorado,  presenting  a  curious 
phase  of  adaptation  in  a  contrary  direction,  its  stem  was 
that  of  a  slender  herb,  yet  in  search  for  noiirishmeiit  in  a 
soil  almost  barren,  its  roots  had  extended  themselves  into 
Hbntus  cylinders  so  numerous  as  completely  to  till  a  llour- 
barrel.  Within  the  (uuiservatory,  as  well  as  in  the  gankii 
around  it,  there  was  abounding  insei't  life.  Since  vital 
processes  go  on  at  surfaces,  the  smaller  an  aphis,  midge,  or 
beetle,  the  more  reason,  we  could  see,  had  the  gardener  to 
dread  it,  for  the  more  food,  proi»ortioiiately  to  its  weight, 
would  it  eat.  The  grasshopper,  the  largest  of  common 
Held  insects,  easily  si.rings  to  a  height  of  tli'ee  feet;  its 
strength,  proportionately  so  much  greater  than  that  of 
man,  is  duo  to  its  smallness,  and  therefore  is  exceetled  by 
that  of  creatures  smaller  still. 

Until  witliin  recent  years  the  collectors  and  students  of 
beetles,  grubs,  and   moths  were   commonly  regarded    as 
people  whose  work  was  perhaps  interesting,  but  of  little 
practical  acctmnt.     To-day  the  economic  side  of  entomol- 
ogy amply  justifies  to  practical  men  the  value,  measured 
in  money,  of  the  science.     At  the  Agricultural  Colleges 
and    Experiment    Stations  scattered    throughout    North 
America  hundreds  of  naturalists  are  busy  finding  out  how 
to  combat  the  insect  foes  of  the  gardener  and  the  farmer. 
Prof.  Lintner,  the  State  Kntoinologist  of  New  York,  has 
observed  a  larva  which  in  twenty-four  hours  consumed  riOO 
times  its  original  weight;  and  u  caterpillar  which,  in  its 
progress  to  maturity,  within  thirty  days  increased  in  size 
10,000  times.     In  view  of  the  depredations  of  insects  so 
voracious,  is  it  any  wonder  that  arsenic  and  other  active 
poisons  are  now  part  of  every  intelligent  farmer's  stock-in- 
trade?     Sometimes  the  farmer  employs  these  poisons  in 
warring  upon  very  different  pests,  as,  for  example,  the  rot 


..  ,'v**?«tMff«-*:rTSffr**^ 


rffftlfigt^^'UP^f^'-'Sf^'' 


to  A  CLASS  IN  GfeOMl-TRY. 

wl.ioli  iittackH  tl.o  point. ..-.luo  t(.  II  fmiKi'H  all  tlio  innro 
f.,r.Mi.l.il.l.-  iKM-uuso  oxtiTinoly  n.i..iiU-.     'I'lu'  pl.yM.'mn,  like 
tlu-  ..iil..ni..l..;,'ist„  unnu.l  willi  ii  imwi'rful  iiiirroHi-opo,  tolln 
UH  tlml  M.iiK.to  orKui.iHi.iH  l.iii.K  ii.l-  tl»'  1"""""   ''•""'« 
runHiimptinn.  (liplilliiTiu.  aii<l  .liHciisfH  iM|Uully  h.tk.hs.     It 
W(.ul.l  Hi'iMu  ihut  tl.o  very  KmuUii.'HH  ..f  tl.c«i>  iHiclonu  ilut.-r- 
miiicH  their  nipi.litv  in  i.nillipii.'ati..i..  tl.cir  intciiso  lu-t.vity 
for  m  il.     Ill  lUiotluT  im.l  K-hh  uiiplfiisiiiit  l.raiicli  of  or^runic 
lif,.  tlu.  law  we  aiv  .oMsidiTiii-  eonu'H  witliin  tlu<  purvmw 
of  tiui  iiaUimliHt.     'Hio  late  I'rof.  A.  M.  Mursliall  of  Vic- 
toria Univorrtity,   Mai.dicHtn'.   Ki.-lan.l,  lias  poiiil.MJ  <..it, 
tlmt  IV  cliaiip'  "f  f«"-in  may  in.  nwcssitatiul  l.y  an  iii(Mva«o 
„f  Hi/,0     CiM-taiii   mi-Hlu},M  of  Hit*  Kt'im^   I.imupoMliii  aro 
about  ..no  sixth  of  an  inch  in  length;  with  a  snrfa.r  .•.„,.- 
purativfly  oxtfn.h-il,  they  ro.,uiro   no  special   ..iKans  t..r 
l)roathing,  an.l  their  ho.ih'S  are  quite  HUiooth.     Slu^s  of 
allied  species,  hut  larger,  have  hranchin«  parts  to  breathe 
through,  and  those  often  take  the  foru)  of  fully  dovol..ped 
gills,— a  distinct  advance  in  c.unidcxity  of  stni.'turo. 

Adhesion  as  Depending  upon  Surface. -'J'hc  boys  well 
know  that  in  winter  ashes  strewn  on  icy  streets  prevent 
foot-passengers  from  falling;   thoy  ha.l    more  than  once 
remarked   the  efficacy  of  ii  little  sand  in   enabling  the 
wheels  of  a  locomotive  to  start  on  an  up-grade:  thoy  were 
to  have  another  example  of  adhesion  as  a  valuable  prop- 
erty      In  one  of  the  rooms  of  the  house  wore  two  pine 
tallies,  each  4  feet  by  2,  and  30  inches  high.     One  of  them 
luid  fluffy  canton  flannel  tacked  over  it.  and  upon  this 
flannel  lay  a  well-worn  table-cloth  of  linen,  about  four  foot 
s(,uare.     The  other  table  stood  bare.     On  gradually  pull- 
ing the  cl.)th  off  the  first  table  it  did  not  begin  sliding  to 
the  floor  until  but  one  seventh  remained  on  the  canton- 
flannel  table-top.    On  transferring  the  linen  cloth  to  the 
bare  table,  it  began  to  slide  off  when  as  much  as  one  third 
of  it  rested  upon  the  wooden  surface.    [Fig.  9.]    A  mag- 


A  TLASS  fN   r.F.OMlTRY. 


3t 


I^MiH  all  tlif  inoro 
111'  |iliysiciiin,  liko 
iiiicnmcoi>»s  lolli* 
Id  liniimti   fniiiK! 
iiiilly  Hcrimis.     It 
•m  iHicloriii  iletiT- 
ir  iiitciiso  lu'tivily 
l)raiii'lH>f  orjjuiiii! 
tliiii  thn  piirvinw 
MiirHliall  of  Vic- 
has   (ioImIimI    out 
(•(1  l)y  an  iiicrcaso 
i   Ijiiiiapniitia  aru 
til  a  Hiirfav^e  i'<im- 
pecial   orgaiiH   for 
mootli.     fSliijiH  of 
(T  purtH  to  brcatlio 

of  fully  (IcVL'lolKltl 

if  striictnro. 
e.— Tlic  boys  well 
vy  strocta  provont 
[    more  than  oucio 
i  in   enabling  tbo 
p-grado:  tlioy  wore 
»  a  valuablo  prop- 
iiae  were  two  pine 
ligh.     One  of  them 
it,  and  upon  this 
len,  about  four  foot 
On  frradually  pull- 
ot  begin  sliding  to 
uod  on  the  canton- 
3  linen  cloth  to  the 
3  nmch  as  one  third 
[Fig.  9.]    A  niag- 


nifying-gl«li  wuh  tbon  laM   .l|»n..   the   'i'";'';.  «'"•;;; 7',;,',';^ 
boys  bow  «xteu.»ivo  the  urea  of  it«llbn"=».     [big.  10. J     1  bo 


71^.9. 


Fig.  10. 


experiment  with  the  tables  was  repeated  by  all  three  boys. 
Thev  th.is  gained  Hon.e  idea  of  adhenion  between  surfaees 
of  Considerable  length  and  breadlh,  and  thereby  appre- 
bended  an  in.p..rtant  aim  ..f  textile  industry .-nanudy. 
giving  adhesion  its  fullest  play  by  si.inning  and  weaving 
threads  of  the  utmost  possible  surfaee.  Tbey  were  mde.l 
in  un.lerstanding  this  by  iindlng  how  mueh  hanler  it  was 
to  pull  out  a  thread  from  a  skein  of  sdk  than  to  with- 
draw a  Htran.l  from  a  sin.ilar  skein  of  thick  twine;  while 
to  pluck  out  a  mameiit  but  two  inches  h.ng  from  its  pla.-o 
in  the  table-cloth  i)roved  an  impossible  task. 

Motion  Thermal,  Electric,  and  Chemic.-At  some  dis- 
tance  from  home,  on  the  banks  of  the  T.ud.ine  (  anal, 
there  stood  a  mill  for  the  manufacture  otdo<.rs,  w.iub.w. 
Bashes,  and  other  articles  for  the  house-bnihler.  AVe  paid 
this  mill  a  visit,  and  took  occasion  to  observe  how  the 
motion  of  the  piston-rod,  doliverod  to  the  tly-wheel,  ^^as 


-.»B«i«aBitr."--«!»s»t 


j4  A  CLASS  IN   GEOMETRY. 

thcnoe  diversified  in  the  various  departments  of  the  factory. 
Here  it  appear  d  as  the  rotation  of  a  planr„g-n.achine, 
there  as  the  swift  to-and-fro  of  a  gang  of  saws,  in  another 
c,„arter  as  tlie  continuous  movement  in  a  !;"«' "r/l'^; 
',ow  curved,  of  a  band-saw;  elsewhere  it  took  the  form  of 
crew-like  advance  in  a  boring  tool,  or  of  a  volute  in  a 
cutter  executing  an  ornamental  design.     The  boys  soon 
comprehended   the  ease  with  which  suitable  appliances 
produce  any  form  of  motion  from  any  otl-r.     1  -Bhed 
hem  to  grasp  something  more:  so  as  we  wei;e  leaving  the 
factory  jtd  I  pointed  to  a  long  jet  of  condensed  steam 
risinc.  to  the  sky  from  an  escape-pipe.     Steam,  I  explained 
.vhen  raised  at  high  pressure,  moves,  as  particles,  with 
great  violence;  hence  two  or  three  pounds  of  steam  stnk- 
ing  against  a  piston  hurl  it  forward  with  tremendous  force: 
when  freed,  as  at  an  escape-pipe,  the  particles  rush  forth, 
at  first  with  the  full  pace  of  their  interior  bombardment 
-the  short,  interrupted  paths  of  their  motion  when  shut 
in,  becoming  a  long  and  straight  line  such  as  that  of  the 
steam-iet  before  us. 

My  pupils  already  knew  that  the  motion  of  a  hammer 
directed  upon  an  anvil  soon  brings  both  hammer  and  anvi 
to  a  warmth  unpleasant  to  the  touch:  reminding  them  of 
this  I  told  them  that  heat  is  the  rapid  motion  of  a  body  in 
particles  too  small  for  detection  even  with  the  microscope. 
«  Could  you  imagine  a  bag  of  marbles  flung  from  one  side 
of  a  field  to  another,  and  then  imagine  each  marble  in 
swift  rotation  while  the  bag  as  a  whole  was  at  rest? 
Thev  could.    Then  they  had  grasped  the  distinction  be- 
tween the"  motion  of  a  mass  as  a  mass,  and  the  motion  of 
its  parts  as  parts;  between  what,  at  a  later  stage  of  in- 
struction, they  would  distinguish  as  molar  and  molecular 
motion.     We  had  now  come  to  a  point  of  uncommon  sig- 
nificance in  our  lessons,  and  I  dwelt  upon  it  with  inten 
that  it  might  be  fully  comprehended.    I  had  the  youngest 


A   CLASS   IN    GHOMUIKY. 


23 


its  of  the  factory, 
plani.ig-miichine, 

saws,  in  another 
line,  now  straight, 
took  the  form  of 

of  a  volute  in  a 

The  boys  soon 

litable  appliances 

other.  I  wished 
B  were  leaving  the 

condensed  steam 
team,  I  explained, 
a8  particles,  with 
ids  of  steam  strik- 
tremendous  force: 
irticles  rush  forth, 
■ior  bombardment, 
motion  when  shut 
Lich  as  that  of  the 

>tion  of  a  hammer 
hammer  and  anvil 
reminding  them  of 
notion  of  a  body  in 
ith  the  microscope, 
lung  from  one  side 
lie  each  marble  in 
lole  was  at  rest?" 
the  distinction  be- 
,  and  the  motion  of 
1  later  stage  of  in- 
lolar  and  molecular 
t  of  uncommon  sig- 
upon  it  with  intent 
I  had  the  youngest 


,,oy  nnd  the  abm.us  of  hi.  early  years,  and  ^^^l^;^;;";^ 
Zg  the  length  of  a  h.ge  table  an     t^^ 

,,,ads  on  their  wuvs:    he  Ihu.  '"'f '     '^  '  ,  ;,,  t,„.owing 

All  these  motions,  as  you  will  du  >  '^'»"  .'^  ; 

the  '"''i^*^'";^  ,  ^,i,„,  n^astered  economical  plans  of 

over,  are  tar  iiom  naNui„  „fi,„,. .  Hms 

orbits  in  which  the  atoms  move  have  ^^  ^   .'     ^'^f  f.^^.^. 

•     1      o  fi,o  ili'imoter  of  a  molecule,     iiotcssoi  .jum' 

,.(11,^  c-irtl.     Tlic  momentum  o!  such  a  fall  "oulcl  ue 
„(  the  c.uin.     '  Vct  an  equiil  momentum  quietly 

c„n,M,.e  to  fo,m  9  poun.U  of  « "«•■  J';"      '  ;,,,       ,„„, 

-1— ;:,x,frr;z^;..f  ^ 


24 


A   CLASS   IN    GEOMETRY. 


prove  tlie  corrtu^tiioss  of  Professor  'J'yndiiirs  n.jures,  is  to 
observe  llie  distiuicc  throiigli  which  a  hammer  falls  in 
desoemliiig  to  an  anvil,  anil  compare  the  heat  of  its  colli- 
sion with  the  heat  set  free  when  hydrogen  and  oxygen 
unite  to  form  water.  Every  common  gas-Uame,  indeed, 
bears  witness  to  collisions  at  velocities  much  greater  than 
that  of  a  musket-ball— collisions  which  bring  the  molecules 
of  gas  to  glowing  heat." 

"  You  are  well  aware  that  a  grindstone  or  a  fly-wheel  will 
burst  by  centrifugal  force  if  too  swiftly  turned;  why  is  it 
that  the  atom,  part  of  whose  motion  we  have  reason  to  be- 
lieve is  rotary,  can  turn  immensely  faster  than  a  grind- 
stone or  a  fly-wheel  ?"  To  answer  this  question  1  showed 
them  a  model  consisting  of  two  wooden  wheels,  each  G 
inches  in  diameter  atid  eacli  covered  with  a  rubber  band 
to  increase  adhesion;  l)oth  wheels,  in  contact,  were  fas- 
tened to  a  frame.  Heside  them  was  a  wheel  12  inches  in 
diameter.  All  three  wheels  were  1  inch  in  thickness. 
[Fig.  11.]     The  tendency  whereby  each  wheel  tended  to 


•Fj.l/. 


fly  to  pieces  on  swift  rotation  was  ascertained  by  mul- 
tii)lying  its  mass  by  the  square  of  its  circumferential 
velocity,  and  dividing  by  the  radius  of  the.  wheel.  As 
C  had  4  times  the  mass  and  only  twice  the  radius  of  A, 


A  CLASS   IN   GEOMirrRY. 


95 


ill's  fi.!;Hres,  is  to 
lainmer  fulls  in 
heat  of  its  coUi- 
gcii  and  oxygen 
as-lhune,  indeed, 
!oh  greater  than 
nn  the  molecules 

)r  a  fly-wheel  will 
urued;  why  is  it 
live  reason  to  be- 
er than  a  grind- 
uestion  1  showed 
I  wheels,  each  G 
h  a  rubber  band 
ontaot,  were  fas- 
leel  1^  inches  in 
ch  in  thickness, 
wheel  tended  to 


;ertained  by  mul- 

ts  circumferential 

if  the.  wheel.    As 

the  radius  of  A, 


the  safe  limit  of  speed  for  .1  was  1.41  (the  square  root  of 
2)  times  that  for  C.     It  thus  became  perfectly  plain  that 
if  a  wheel  were  as  minute  as  an  atom,  it  could  safely 
withstand  a  marvellous  swiftness  of  rotary  motion.     Our 
model    tau-ht    us    more.     The    peripheries    of    the    two 
small  wheels  in  the  course  of  one  rotation  described  two 
circles  whose   united   length  e(iualled    that   of   the   orbit 
described  in  ono  rotation  of  the  rim  of  the  largo  wheel,— 
and  this  while  A  and  />'  had  together  l)ut  half  the  mass  of 
a     Again  imagining  the  small  wheels  reduced  to  atomic 
proportions,  it  was  evident  that  in  the  transmission  of 
motion  by  superficial  adhesion,  which  may  be  the  way  in 
which  electricity  is  transmitted,  we  '.    re  helped  in  trying 
to  understand  the  surpassing  velocity  of  electrical  conduc- 
tion.    Were  an  inch  cube  to  be  filled  with  atoms  such  as 
those  of  Sir  William  Thomson's  estimate,  the  elde  t  boy 
calculated  that  in  comparison  with  the  iiudi  cube  they 
would  have,  if  cul,ical,  u  total  superticies  ;(iO,Ut)0,tHM)  times 
as  great,  or  a  surface  of  almost  \\  sfiuare  miles.     I'he  law 
of  surfaces  and  volumes  had  taught  us  much  with  regard 
to  masses;  it  brought  us  at  last  to  the  consequences  in- 
volved in  the  infinitesimal  dimensions  of  the  units  which 
form  masses,— their  astonishing  capacity  for  motion,  the 
amazing  velocity  whereat  they  can  propagate  motion. 
The  Cube  and  Pyramid;  the  Sphere,  Cylinder,  and 

Cone Our  next  lesson  was  intended  to  bring  out  the 

relations  which  subsist  between  several  of  the  principal 
forms  of  solids.  Two  series  of  models  in  wood  were  ac- 
cordingly obtained.  The  first  consisted  of  a  cube  having 
a  base  5  inches  square,  and  a  wedge  and  pyramid  of  similar 
base  and  height.  [Fig.  Iti.]  T'he  second  series  comprised 
a  sphere  5  inches  in  diameter,  and  a  cylinder  and  a  cone 
each  5  inches  in  breadth  and  height.  [Fig.  13.]  We 
took  up  the  first  series,  when  a  moment's  comparison  of 
the  sides  of  the  wedge  and  cube  told  us  that  one  coutuined 


26  A  CLASS  IN   GUOMKTRY. 

half  as  nuich  wood  as  the  other;  but  tliut  th.  pyramid 
contained  a  third  .s  tnuch  as  the  cube  was  not  evident  on 


Tliis  relation  was  l)r()U<rlit  out  in  weigh- 
i;  but 


•Sl^the  pynmad  a^^'^^ibe;  but  a  n.ore  satisfactory  demon- 


stration was  desirable,  for  what  was  to  assure  us  that  the 
two  solids  were  of  the  same  ^pecnfic  ^-rav.ty  .^     N^^now^ 


T.^.lf 


took  a  clear  glass  jar  of  cylindncal  "^7'^;' "jr^jt'^Suh 
inches  in  width  by  10  in  height,  aud  half  hUed  it  Mth 


A  CLASS  IN   GEOMETRY. 


47 


lilt  the  pyramid 
8  not  cviiloiit  on 


rht  out  ill  wcigh- 
tisfactory  tlenion- 


,ssure  lis  tliat  the 
ravitv?     We  now 


orinr.  moasuring  7^ 
half  tilled  it  with 


water.    The  models,  duly  varnished,  and  1  de^.  .    h  Um 
so  as  to  sink,  were  then  suecess.u-ly  numer.  d,  ami  I 
displaeement  of  the  water  noted  with  the  aid  o    a  fc 
rule.     This  proved   that  the  pyranud  had  one  th.  d    he 
vol  me  of  the  cube,  that  the  same  proportion  subsisted  bo- 


'Tig  15- 

tween  the  cone  and  the  cylinder,  and  that  the  sphere  had 
twice  the  contents  of  the  cone.     [Figs.  U  and  b,.J 

A  Square-root  Extractor.-In  a  lesson  given  sho.tly 
after  L  formation  of  our  little  class  (p.  «)  it  had  been 
n^ ule  clear  that  plane  figures  of  the  same  form  Inue  areas 
"      n<^asthe  square  of  their  like  dimensions     Dividing 
or  wedge  at  a  Iriangular  side  by  three  parallel  lines,  an 
eq  aT  dfstance  apart  [Fig.  16],  I  asked  how  the  area  0 
SI  smallest  triangle,  IfJC,  so  laid  off  and  t  -   of  the  n  x 
smallest,  FGC,  would  compare  with    he  ar  a  of    he  large 
trian.rle  formed  by  the  whole  side  of  the  wedge,  A  EC.      As 
h   square  of  their  sides,"  was  the  answer,  the  correctnes  o 
hi  h  was  there  and  then  confirmed.     Dipping   he  we.lge 
b   ow  the  surface  of  the  water  in  the  jar,  edge  downwa. 
'v    observed  the  water  displaced  as  the  square  of  the  depth 
of  immersion.    When  we  reversed  the  process,  the  wedge  be- 


98 


A   CLASS   IN   (;K()MI:TRY. 


ciiniP  a  simple  iiioaiis  of  oxtriiotin,tr  sf|iiiirc  root.     We  hail 

ulioiidy  (liviiltMl  the  wodge  into 
four  parts  l)y  cciuicHstant  par- 
allel lines;  we  now  divided 
the  vertical  i)lay  of  its  dis- 
placement into  sixteen  eqnal 
parts  marked  alonjf  the  side 
of  the  jar.  'I'hen,  for  example, 
if  we  sought  the  sriiiure  root 
of  y,  we  immersed  the  wed<?e 
with  its  edt^e  downward  until 
it  had  displaced  water  to  line 
9,  on  the  side  of  the  jar ;  on 
_  the  wedge   the  water  stood  at 

^'  line  15,  the  square  root  of  9. 

A  Cube-root  Extractor.— In  one  of  our  llrst  lessons  wo 

had  learned  (p.  .S)  that  solids  of  the  same  form  vary  in 

contents  as  the  cuhe  of  their  like  dimensions.     New  proof 

of  tliis  rule  also  was  at  hand.     We  took  a  cone,  and  mark- 
ing it  otf  into  three  sections  of  equal  breadth,  we  had  within 

its  surface  three  cones:   lirst,  the 

whole    figure,    ACfD;   second,  a 

cone     two     thirds    the     height, 

breadth,  and  depth  of  the  whole 

figure,  AhV;   third,  a  cone  one  ^7 

third    the    height,  breadth,  ai\d 

dei)th  of  the  whole  figure,  AFB. 

All  three  were  in  strictness  solids 

of  the  same  form,  so  that   their 

contents  were  in  the  ratios  of  1,  8, 

and  27— the  cubes,  respectively,  of 

1,  3,  and  3.     The  cone,  as  a  whole* 

apex  downward,  we  now  immersed 

in  the  jar.      It  Avas  oh  erved  to 

displace  water  as  the  cube  of  its  dei)th  of  immersion, 


"OWtfWW^'-i'W"'  • 


J 


A  CLASS  IN   GF.OMF.TRY. 


39 


re  root,     We  liad 
(1  the  woilge  into 

oquiilistant  piir- 
ive  now  divided 
])lay  of  its  dis- 
to    sixteen   equal 

alon}?  the    side 
lien,  for  example, 

the  sfiHure  root 
lersed  the  wed<jo 

downward  until 
;ed  water  to  line 
3  of  the  jar ;  on 
le  water  stood  at 
are  root  of  9. 
ir  first  lessons  wo 
mie  form  vary  in 
lions.  New  proof 
11  oone,  and  inark- 
1th,  we  had  within 


iTT^.ia 


37' 


)lh  of  inuucrsion, 


and  therefore  could  be  impressed  into  the  sorviec  of  ex- 
tracting cnhe  root.     For  this  purpose  its  total  play  of  dis- 
placement in  a  jar  r.,l  inches  interior  diameter  was  divided 
into   '•;   e.iual  parts;    the  cone  wus  already  >''n>-ked   off 
into  three  sections  of  eciual  breadth.     [Kii,'.  1  <•]     !«  «"'<» 
the  cube  root  of  S,  we  lowered  the  (■one,  apex  dowiiward, 
until  the  water-level  was  brou.u;ht  to  S  on 
the- side  of  the  jar;   at  that   moment  the 
li(iuid  encircled  the  cone  at  section  '-!,  the 
cube  root  of  8.     The  pyramid  immersed  in 

the  larger  jar  was  also  found  to  be  avail- 

al)le  as  a  cube-root  extractor.     After  our 

first  experiments  in  extracting  square  and 

cube   root    by   immersion,   we    found    it 

advantageous  to  use  as  tall  glass  jars  as    8^ 

we  could  get;  the  difference  between  high 

and  low  tide  being  thereby  increased,  we 

could  make  our  scale  on  the  side  of  the 

jar  much  longer,  and  therefore  each   of 

its  divisions  was  more  clear  to  the  eye. 

We   also  found   it   helpful   to  have   the 

cone  descend  from  a  rod  turning  in  the 
screw-thread  of  a  light  framework.     [Fig. 

18.1 

Perspective. — Measuring  the  cone  and 
pyramid  at  each  of  their  sectional  divis- 
ions, the  boys  were  required  to  ascertain 
the  rule  governing  their  increase  of  sec- 
tional area,  arriving  at  the-  old  familiar 
law  of  squares— a  law  true  not  only  of  all 
solids  converging  regularly  to  a  point,  but 
of  all  forces  divergent  or  radiant  from  a 
centre,  simply  because  it  is  a  law  of  space  through  which 
such  forces  exert  themselves.  That  the  cone  of  our  ex- 
periments  represented  the  transmission  of  light  from  a 


30 


A  CLASS  IN   GEOMETRY. 


circle  was  obsorved  by  one  of  the  boys  during  the  pro- 
joiition  of  some  plvotognipbs  of  seonory  tiirough  ii  stt-rooi.- 
lioon  ill  u  publio  hall,  where  ho  drew  my  attention  to  the 
cones  of  rays  shot  forth  from  the  exhibitor's  lantern  in 
the  fTMllcry— ruvs  clearly  aeHm'd  in  the  dii«t  of  the  air. 
More  Ihaii  oiwe  we  had  all  noticed  the  beautiful  eil'ect  iu 
the  skv  known  as  "drawing  water,"  duo  to  solar  beams 
icnecled  and  scattered  by  a  cloud  into  lines  much  resem- 
bling tliosi^  which  bound  a  pyrainid. 

I  hud  a  camera  with  which  I  occasionally  permitted  tho 
eldest  boy  to  take  pictures.  Arranging  on  an  easel  a  sheet 
of  paper,  4  feet  square,  I  had  him  take  its  photograph, 
'{"hen  bringing  the  easel  to  a  point  half  as  far  from  the 
.-uniera  as  at  first,  1  had  him  photograph  a  sheet  of  paper 
'>  feet  square.  Its  picture  was  of  exactly  tho  same  sizo  as 
that  in  the  first  experiment.  I  asked  liim  to  imagine  lines 
drawn  from  both  pieces  of  paper  at  their  respective  dis- 
tances to  their  imago  in  the  camera;  he  soon  saw  that 
these  lines  would  body  forth  a  pyramid.  We  had  here  an 
explanation  of  how  hands  and  feet,  extended  toward  a 
camera  bv  a  sitter,  are  exaggerated  in  a  photograph,  espe- 
cially when  a  common,  simple  lens  is  employed.  From 
this  "we  passed  to  some  elementary  consideration  of  per- 
spective-an  allied  and  timely  theme,  as  tho  boys  had 


Vttni»ljinhj_l\>iij1^ 


Ti4.i9- 


begun  to  practise  drawing.    Taking  a  picture  of  a  box,  I 
showed  how  its  outlines,  as  far  as  they  went,  were  but  short 


A  CLASS   IN    r.KOMFTRY. 


31 


luriii}:';  tlio  pro- 
rough  11  8tt'rooi»- 
iitteiitioii  to  tho 
itor'H  lantern  in 
(lust  of  tliu  air. 
lantil'ni  olTcct  iu 
to  solar  beams 
les  niiuli  resuiii- 

ly  pcrniittud  tho 
,  an  oasol  a  sheet 

its  photograph. 

as  far  from  the 
a  sheet  of  paper 
the  same  size  as 

to  imagine  lines 
ir  respective  dis- 
e  soon  saw  that 

We  had  here  an 
tended  toward  a 
)hotograph,  espe- 
iiiployed.  From 
[deration  of  per- 
ns the  boys  had 


ricture  of  a  box,  I 
int,  were  but  short 


lengths,  or  parts,  of  the  lines  which  boiin.l  a  pyramid  at 

its  angles.     [Kig.  19.] 
Triangles  and  Circles.-We  now  ad.lres.so.l  oursclvs  to 

the  study  of   triangles,  beginning  by  .hawing   ihcn,    on 

paper,   in   acute-,  right-,   and   obtu.e-ang  ed    b.r.ns.      We 

found  that  each  of  these  triangles  nugbt  lu -;''•"<';'« 

half  a  parallelogram  of  ccjual  base  and  bcgbt.  so  that  «.th 

u  foot-rule  the  con.putation  of  a  triangular  '^'•-^ ;';'-^  ^ ^^^ 

We  also  found,  by  our  sector,  that  the  .bree  angle     .         y 

triangle  are  equal  to  two  right,  angles.     Keeping  d.M.n.    v 

,.  mh.d  the  three  types  of  trianglos-tbe  aeute-   .be  r.gbt- 
;:;theobtuse-angled,-we  next  proceeded  to  e.lcu  a  e    ow 
the  square  of  the  longest  side,  or  the  square  of  o,.e  of  U 
equal  sides,  non.pares  with  the  sum  of  the  squares  of  the 

.  tlr  two    ides.     At  the  end  of  a  great  numy  tests  we  eon- 
duJldTl^tonlyinthecaseoftheright-an^odt^^^^^ 

the  square  of  one  side  equal  to  the  sum  of  the  ^'l"-'  ^^« 
the  o?her  two  sides;  and  we  found  that  he  ";;>--;  l^^- 
ueute-  or  obtuse-angled  triangle  ai.proacbes  the  forn.  o  a 
right-an^^led  triangle  the  more  closely  does  it  exemphly 
i  relaUon.  [Fi;  20.]  Formal  proof  of  wnch  relat.or. 
and  of  other  relations  tluU  wo  had  arrived  at,  awaited  tho 

boys,  I  said,  in  the  Euclid  they  were  to  study  in  due  season 
Before  taking  leave,  for  the  time  being,  o    our  tnangles,  I 
had  the  boys  cut  in  stiff  cardboard  a  number  of  flat  frame 
aving  '6, 1  5,  and  more  sides.    Fastening  these  frames  at 
t:: "o'rnirs'wlth  pins,  we  found  the  triangular  h-arne  the 
only  one  not  liable  to  fold  down  and  collapse     L* 'g-^l; 
H  ro  was  a  fact  to  be  remembered,  because  by  and  by  it 
woi  d  help  to  make  clear  why  builders  of  roofs  and  bridge 
chle  foi-  their  trusses  triangular  shapes,  or  the  trapezoidal 
forms  which  share  their  stability. 


-jimniiu  W  m' 


s» 


A   CLASS  IN    C.EOMKTRY. 


Two  iiu'iiiltors  of  my  litllo  class  were  ft1)lo  incntiilly  to 
Bqiiarc  miiiilHirs  ooiitainiiis  tlim^  miim'iuls— by  »)reiikiiig 
n\)  11  iiumbor  into  two  iiiimaj,'etil>lo  party. s(|uariiiK  the  first, 
atldiiiK  to  tliis  twico  tlio  product  of  tlic  first  and  second, 
and  then  adding  the  square  of  tho  second.    Taking  a  sheet 


of  paper  we  drew  a  diagram  in  illustration  of  the  process. 
[Fig.  2'^.]  After  repeating  the  figure  in  outlines  having  a 
base  divided  in  various  proportions,  I  wrote  beneath  the 
illustrations, 

and  explo.ining  how  the  formula  should  bo  read,  the  boys 
began  to  see  that  algebra  bears  much  the  same  relation  to 
geometry  and  arithmetic  that  numerals  do  to  numbers 
written  out  in  words— that  of  an  invaluable  shorthand;  a 
shorthand,  too,  which  expresses  relations  in  their  widest 
generality. 


-WHWIIt-IW 


A  CLASS   IN  C-.F.DMUTRY. 


ii 


ivl)lo  inentiilly  to 
•ul8— by  breaking 
i(|Uiiriiig  the  first, 
first  imd  seooiid, 
.     Tuliiiigiv  slioct 


U 


ion  of  the  process. 
1  outlines  having  a 
wrote  beneath  tlio 


I  bo  read,  the  boys 
le  same  rehition  to 
lis  do  to  numbers 
lable  shorthand;  a 
)us  in  their  widest 


Wliilf  I  wuH  ghid  to  i-niploy  observed  exumiilos  and 
models  in  the  iiistrnetion  of  my  inifils.  I  wislied  them  to 
grusi)  certain  geometrical  relatinns  tiiroiigii  exercise  of  the 
imagination.  Heminding  them  that  any  triangle  nmy  bo 
conridercd  as  half  a  iiarallelogram  of  e<iual  base  and  height, 
wo  proceeded  to  tiie  study  of  the  circle.  1  told  thetn  the 
old  way  of  ascertaining  the  area  of  a  circle  by  conceiving 
it  to  be  made  up  of  an  indelinitely  great  number  of  tri- 


-B- 


angles,  whose  bases  becojno  the  circle's  circumference,  and 
whose  altitude  is  the  cinde's  radius.  IJolling  the  cylindri- 
cal model  round  once  on  a  sheet  of  paper,  we  marked  off 
its  circuit;  this  was  made  the  base-lino  of  a  parallelogram 
having  a  height  equal  to  half  the  cylinder's  breadth;  half 
that  area  was  clearly  equal  to  the  surface  of  the  circle 
forming  the  cylinder's  base.  [Fig.  2:5.]  Another  mode  of 
indicating  the  relation  between  the  circum:  .nee  and  the 
area  of  a  circle  was  followed  by  the  boys  with  fair  prompt- 


A  C.l.ASS  IN   lU-.C)MJITRY. 


no...  ^^^^'U^ ;..i 7;;:,:;::::;^,;:;.;:::^: 


Hupposing  lliu 


bo     Je-tlu.bron,l,tto«ooUu.t.f.o,r.c 

roll  a  rouml   oneo   is  found    to   have  .U41  . 

,n  ul   units   for   its   circun,foron..o,  ,  a  arcu 

^W    ni        be  .7854,  or  one  half  of  one  ha  f   as 

^     n.ucl  cxprcsea  in  Bupcrficial  nnUs  of  the 

game  order.     [Ki«.  2-t.  1  ^     concrete :  1  phiced 

,1  map  of  >lontrtai  a  ^^^^^  ^^^.^^^  ^^  ^.„.,.,o 

l,i«  1,0.^0  a.  a  con  ,    ^^1  J^^^^  ^^^  ^,^,,,  ^,,,  ,, 

''"■•".^''r"       On      «  bt  10  bo  could  ride  thrice  a«  fast 
15  nunutes.     On  lus  "'^J*^  ,^     j^,       r,jj|i„g  equal 

"  Now,"  1  ..ked.  "  I'l^;^  -  ^  "  tiLto."  lie  did  bo, 
to  the  distance  [;;  ^.^tjl  lengthened  thrice,  he  found 
and,  with  the  radn.s  of  Ins  "U    J,  ^^.^  ^^„^. 

that  with  im  wheel  he  had  «;•;';;;;;  J^,^  ^o-day. 
„,and  ninefold  as  -t-;^«^^,  ^^  J  ^.^  pid  transit  in  ex- 
a  similar  object-less.J  -  *^- -^^^  ^,,,  Jig,,  „,a  to  the 
tending   the  area  o     '^J^y";    .  j.e  which    employs 


A   CLASS   IN   r.KOMKTRY. 


$$ 


•  (link  to  l»o  iiiii'lo 

eoiuTiilrif  rinHH- 

ter,  uiul  each  ring 


lii-i'il,  "How  iimiiy 
y— half  II  iiiillion." 
,!mt  wtuiltl  bo  tlie 
fre  ring?"    "  Half 
niH  the  ri'ply.    'l'l>o 
,  800  that  if  a  circle 
11(1    to   have  ;J.141() 
iinforciico,  its  area 
alf  of  one  half  as 
rtieial   units  of  tho 

cs  concrete;  1  placed 
01,  the  table.     With 
then  tlrew  a  circle 
nco  he  conlil  walk  in 
I  ride  thrico  as  fast. 
B  with  a  radius  equal 
linutes."     Ho  did  so, 
ened  thrice,  he  found 
f  country  at  his  com- 
went  afoot.    To-day. 
if  rapid  transit  in  ex- 
8   might  add  to  the 
rvice  which    employs 
m  that  of  the  horse,— 
I  of  safety. 


T.^.X5-. 


Conical  Surfaces.  'rbi>  bovH  well  knew  that  in  dividing 
a  |>it'  by  lini'H  niiiniiij,'  fmni  it.-*  cciilrti  carji  piece  iw  jnut,  an 
lar^'c  as  liie  inclmlcd  arc  of  tlic 
cintlc  niaki'H  it.  Witii  tiiin  fact 
in  their  iiiinds  1  asked  tlieiii  to 
lay  a  cone  on  its  side  and  ml!  it 
round  once  on  a  slieet  of  paper. 
'I'iie  cone  in  its  excursion,  ilnly 
nnirked  ont,  jiave  a  sector  ol'  a 
(fircle,  wilicll,  like  the  piece  ol" 
pi(^  it  reseinl)lcil,  had  an  area 
proporl  ioncd  to  the  leii;r|h  of 
ils  arc.  On  a  former  occasion 
we  had  found  that  tlie  area  of 
a  cinrle  is  its  eirciiinfiM'enco 
innitiplied  by  iuilf  its  radius: 
thertd'ore  the  area  of  tho  sector 
before  us  was  tiio  length  of 
ils  arc  niiiltipIi(Ml  by  half  its 
radius.  Hecallin.L;  that  this 
sc(!tor  was  e(pnd  to  tho  curved  surface  of  the  cone  which 
in  rolling  had  descril)ed  it,  tho  boys  saw  tiiu  reason  for  the 
rule  employed  in  ascertain  in  jj;  tho  curved  surface  of  a  (joiio 
— tho  multiplication  of  tho  circumference  of  its  base  (tho 
arc  of  our  sector)  by  half  tho  slant  height  (half  tho  radius 
of  our  sector).     fKig.  ^i"). | 

Great-circle  Sailing.— At  this  stage  in  the  course  of  our 
lessons  there  was  much  public  interest  in  tho  (irecly  ex- 
podiliou  to  tho  North  Pole:  I  thought  that  I  would  lead 
up  to  a  consideration  of  that  Arctic  voyage  by  an  illustra- 
tion of  great-circle  sailing.  Turning  a  terrestrial  globe 
upon  its  axis,  wo  observed  that  tlie  Gillolo  Islands  and 
('ape  San  Francisco  occupied  points  on  the  equator.  A 
ship's  shortest  course  plainly  lay  tdong  the  equatorial 
liuo  which  joiued  them.    I  asked  whicJi  wus  the  shortest 


gfi  A  CLASS   IN   GEOMETRY. 

route  from  Portland,  Oregon,  to  the  northern  extremity  of 
route  iiuiii  i  concurred  in  a 

t„e  J«r;;;--."\';"^,,  4\r,  J!:"  „t-„orther„  latitnJo 
4^    ,10  ,    tl"-  P'"  '  di»nirtci-  to  the  cqiiutor,  alul 

Z;\,.r.,,oH.t  route  w™UU.ca^ 

CaZ:;^t  o-,Ui„gt„e  crtl,  i.  a  pUu.e  iuchnod 


to  the  googtaphica.  e,,,.;:;:  [Pi^  ^;^J'^ZZ 
ceeded  to  measure,  lo  arcs  «'  !  °'\'  ™*^^^^^^^  itp,„,ea 
f;:!trirr  ;rt::  r:Se';reLr  M  „tre». 
or  NeVyork  to  Liverpool,  sin.ply  beeause  the  sphere  ot 


^"(Bft^gr^S^ 


■.;Jii^wj.--eaJtJ^'J^igtiJ^J^i-^^'^^ 


A   CLASS   IN   GEOMETRY. 


37 


tliern  extremity  of 
ys  concurred  in  a 

nortliern  latitude 
"  Taking  a  brass- 
to  the  equator,  and 
ho  lads  saw  at  onco 
p  somewhat  toward 
age,  and  somewhat 
hat  if  two  ports  are 
e  of  which  that  arc 

track,  because  it  is 
•aiglit  line :  f urtlier- 
practically  no  other 
iu  a  plane  inclined 


Ti^Rfc. 


;.  26.]  We  now  pvo- 
t  circles,  the  distance 
to  Liverpool:  it  proved 
.  from  either  Montreal 
because  the  sphere  of 


the  earth  contracts  more  and  more  as  its  polos  are  ap- 
proached. Now  we  could  understand  how  tlio  possibility 
of  finding  an  open  polar  sea  dazzles  the  imagination  of  ex- 
plorers, and  lures  them  one  after  another  to  scenes  of  priva- 
tion and  death.  Wore  navigation  })roved  to  be  free  across 
the  Arctic  circle,  a  new  an<l  nuich  shortened  route  would 
at  ouco  be  laid  down  between  Asia  and  Europe,  and  Asia 
and  tlie  eastern  coast  of  America.  I  was  careful  to  point 
out  that  the  navigator,  like  hi,-,  friend  the  builder  asliore, 
has  to  consider  something  besides  the  immediate  indica- 
tions of  geometry.  An  ocean  route  which,  at  first  view, 
seems  most  circuitous,  is  really  the  best  when  by  choosing 
it  a  captain  avoids  adverse  wii\ds  aiul  currents,  shoals  and 
reefs,  or  the  chance  of  meeting  ice-floes  and  icel)ergs. 

(The  New  York  Central  and  Hudson  Kiver  Kailroad  de- 
flects considerably  from  a  straight  line  drawn  between  its 
terminal  points,  New  York  and  Buffalo;  it  is  because  of 
that  deflection  that  the  road  enjoys  its  economy  in  work- 
ing expenses.     It  is  better  for  a  railroad  to  make  a  mod- 
erate detour  than  to  climb  a  hill  for  the  sake  of  following 
a  direct  line;   in  conforming  to  the  valleys  of  the  Hud- 
son and  the  Mohawk  rivers  the  railroad  named  has  easier 
gradients  than  those  of  any  other  line  running  westward 
from  the  Atlantic  seaboard.     Incitlentally,  too,  the  roail, 
in  passing  through  a  succession  of  towns  and  cities  origi- 
nally founded  on  the  Hudson  and  Mohawk  rivers,  has  a 
business  vastly  larger  than  if  it  had  been  built  in  a  straight 
line.     This  advantage  in  apparent  indirectness  is  shared 
by  nearly  every  other  important  railroad  in  America.     In 
Russia,  where  military  considerations  outweigh  those  of 
trade  and  commerce,  several  of  the  principal  railroads  are 
as  nearly  straight  as  their  engineering  difliculties  would 

permit.) 

The  Reading  of  Plans — Not  many  years  ago   there 
dwelt  in  Albany,  New  York,  a  stove-founder,  who  had  to 


38 


A  CLASS  IN  GEOMETRY. 


waste  ...any  tl,ousa..a8  of  dollars  bcoansc  ho   could   no 
r.  'h.o  f.'...  a  drawing  how  the  stove  it  dop.cte.l  would 
^^  r     lu  every  case  of  a  ,.ew  design  it  was  neeessary  to  go 
to    he  la.-ge  Jxpcsc  of  building  hi.u  au  actual  pattern, 


and  these"patterns  had  not  seldom  to  be  rejected  as  un- 

suitable.     That  .ny  pupils  .night,  if  possible  be  spa  ed  h.a 

n'citv,  they  were  give.,  a  variety  of  exere.ses  wUh  sol.ds 

lud  their  «eoUonal  planes.    Our  5-mch  models  of  the 


111 


i;:^^:E2E^S®B9SSB)l^r" 


A  CLASS   IN  GEOMETRY. 


39 


use  he  could  not. 
it  depicted  would 
viis  necessary  to  go 
uu  actual  pattern, 


to  be  rejected  as  un- 
wssible,  l)e  sjjured  his 
)f  exercises  with  solids 
i-iucU  models  of  the 


sphere,  cylinder,  and  cone  were  each  sawn  in  halves.     By 
applying  the  cleft  surface  of  each  to  paper,  and  having  the 
resulting  figure  outlined,  I  made  intelligible  the  genera- 
tion of  solids  by  the  revolution  of  planes.     The  square, 
triangular,  and  circular  outlines  were  then  cut  out  of  the 
paper;  each  was  divided  by  an  axis,  aiid  made  to  revolve 
around  it.     [Fig.  27.]     Procuring  a  variety  of  lathe-work 
from  a  carpenter's  shop,— hand-rails,  table-legs,  and  the 
like,— the  boys  drew  them  in  section,  and  these  drawings 
were  then  compared  with  the  woodwork  sawn  down  its 
centre.     In  the  main  the  accuracy  of  the  drawings,  a "ter 
the  first  attcmi)ts,  was  commendable.     A  school  i)rovided 
with  the  tools  for  manual  training  could,  of  course,  have 
much   developed   this  elementary  education  of   the  eye. 
Without  such  a  school  the  boys  with  a  little  practice  were 
able  to  make  plans  of  the  houses  they  lived  in,  and  to  read 
similar  plans  published  in  a  popular  journal  of  architec- 
ture.    Through  hu-k  of   this  easily  cultivated  power  of 
being  able  to  body  forth  how  a  house  will  look  from  its 
drawings,  there  is  much  endurance  of  discomfort,  or  a 
wasteful  alteration  of  rooms,  halls,  and  staircases,  by  people 
wb ')  have  neglected  to  make  their  eyes  stereoscopic,  even 
in  a  rudimentary  degree. 

Proof  of  a  Newtonian  Theorem — One  evening  all  the 
lights  in  our  sitting-room  were  extinguished  excei)t  that  of 
a  small  lamp.  This  lamp  I  placed  in  various  positions,— 
in  the  centre  of  the  room,  near  a  wall,  in  a  corner  near  the 
floor,  and  so  on.  At  each  place  of  momentary  rest  I  asked 
the  lads  to  imagine  the  flame  bisected  by  any  plane  so  ex- 
tended as  to  divide  the  roojn  into  two  parts.  They  were 
brought  to  see  that  no  matter  how  small  one  of  these  parts 
might  be,  even  if  only  a  square  foot  or  so  in  a  corner  near 
the  floor,  it  received  half  the  flame's  light,  because  the 
bisecting  plane  sent  half  the  rays  into  that  little  bit  of 
corner.    For  the  sake  of  illustration  we  assumed  the  area 


40  A  CLASS  IN  GEOMETRY. 

of  the  bit  of  corner  to  be  ,U  «>«  ^^^^  ^\  ^he  r^^i^-^^  ^* 
III  room.     As  both  these  areas  received  equal  quantities 

0  light,  t  was  evident  that  the  geometrical  mean  distance 
of    he   arger  section  from  the  flame  was  10  times  that  of 

1  snU  er.     When  the  boys  had  mastered  this  thoroughly 
they    aw  that  nothing  was  changed  if  we  imagined  the 

amp-flame  reduced  to  the  dimensions  of  a  luminous  par- 


Ti^.^b. 


tide  and  that  we  were  now  free  from  having  to  disregard 
h  sh  dows  cast  by  the  body  of  the  lamp.  But  suppose 
a  radical  change  of  affairs,-the  walls,  floor,  and  ceiling  to 
rad  a  e  light  of  uniform  brilliancy,  on  a  particle  otherwis 
dark  would  the  particle  receive  on  any  two  halves  equal 
amounrs  of  lightf  It  took  a  good  -hUe  for  U^  boys  to 
understand  this  case  as  the  converse  of  the  other,-to  see 
thatTaccording  to  this  new  supposition,  if  a  plane  divided 


0 


f  the  remainder  of 
a  equal  quantities 
ical  mean  distance 
18  10  times  that  of 
3d  this  thoroughly, 
:  we  imagined  the 
of  a  luminous  par- 


1  having  to  disregard 
1  lamp.  But  suppose 
1,  floor,  and  ceiling  to 
[1  a  particle  otherwise 
any  two  halves  equal 
while  for  the  boys  to 
3  of  the  other,— to  see 
ion,  if  a  plane  divided 


A  CLASS  IN   GEOMETRY. 


4t 


the  room  into  parts  having  areas  as  1  to  100,  then  such 

Ine  .<.uld  perrorce  bisect  the  particle  a    a  geo.net    ca 
mean  distance  of  10  from  the  larger  section    and  at  a 
gcon.etricul  mean  distance  of  1  fron.  the  s.nallor  secUon; 
U.at  the  particle  when  shone  upon,  instead  of  bon.g  raduu    , 
would  r  ceive  on  any  two  of  its  halves  prec.sely  e.ju 
quantities  of  light.     1  now  told  the  class  that  we  had 
seized  the  substance  of  one  of  Newton's  famous  proposi- 
t bns.     In  his  "  Principia"  he  proves  that  if  w.tlnn  a  sphon- 
cal  shell  of  infinite  thinness  a  particle  is  placed,  which 
particle  is  attracted  by  the  shell  inversely  as  tl-  ^q"!"-  o^ 
Uie  distance,  then  at  whatever  point  it  may  be  put  tl^ 
particle  will  remain  at  rest,  the  attractions  on  any  two  ^ 
fts  halves  exactly  balancing.     Our  luminous  particle  had 
not  only  showed' this  proposition  to  be  true  of  sphencal 
shells,  but  of  shells-interior  surfaces-of  any  form  what- 
ever, provided  they  have  no  re-entrant  angles  relatn^ly  to 
the  particle.    Indeed,  any  shell  not  spherica    may  be  re- 
garded a«  made  up  of  points  in  a  series  of  spherical  she  s 
expanding  out  from  an  enclosed  particle  as  a  centre,-the 
po  nts  being  exposed  to  form  the  continuous  surface  pre- 
ronted  to  the  particle.    All  this,  I  told  the  boys,  would 
apply  when  they  took  up  the  study  of  light  and  heat,  and 
can!  I  the  theory  of  exchanges.     They  were  now  ready 
to  advance  another  step  in  the  use  of  light  as  a  means  of 
making  visible  certain  laws  of  geometry. 

A  Sphere  and  its  Enclosing  Cylinder.-I  asked  them  to 
imagine  a  spherical  shell  lighted  by  a  particle  at  its  centre. 
iTey  apprehended  at  once  that  the  shell  would  be  uni- 
formly Uluminated.    "  Now,"  I  said,  "  suppose  the  part  cl 
replaced  by  an  axis  evenly  shedding  forth  beams  equa    i 
Zount,but  in  directions  at  right  angles  to  itself,    -shell 
will  receive  exactly  as  much  light  as  before  will  it  no  ^ 
"  Of  course."    "  Now  let  us  suppose  the  spherical  shell  to 
be  transparent,  and  enclosed  by  a  cylindrical  one  having 


^  A  CLASS  IN  GEOMETRY. 

the  same  height  and  breadth,  will  not  the  cylinder  also  be 
Uniformly  illLinated  throughout?"  "Yes."  "And  with 
Z  same  brightness  as  the  sphere  in  our  firs  suppos  - 
tion?"  "Yes;  tlie  radii  will  have  equal  length  in  both 
cases  "  "  Well,  then,  the  curved  surface  of  that  cylinder 
rr  have  precisely  the  same  area  as  the  sphen,  for  both 
receive  an  identical  amount  and  brightness  ofhghL      Kot 


80  promptly  came  the  con.ment,  "  So  it  must  "  [V\g.  29.] 
I  sdd  to  the  boys  that  if  they  kept  all  tins  m  mind  it 
rould  bv  and  by  enable  them  to  understand  how  the 
curved  surface  of  the  earth,  globular  as  it  ,s,  can  be  repre- 
sented on  the  flat  surfaces  of  maps  and  charts. 

Here  let  me  say.  parenthetically,  that  an  important  point 
has  since  occurred  to  me  in  connection  with  tha   le^on- 
tho  sphere  in  the  second  supposition  is  uniformly  illumi- 
Lted  from  its  axis  exactly  as  when  the  light  streams  ^r  h 
in  all  directions  from  its  centie.     iluis 
tliere  is  new  proof  of  a  familiar  method 
of  finding  the  area  of  a  spherical  zone  or 
segment- the  enclosing  a  sphere  within 
a  vertical  cylinder  and  producing  the 
parallel  horizontal  planes  which  bound 
the  segment  until  they  cut  off  a  section 
of  the  cylinder;  the  area  of  that  cylin- 
drical section  is  obviously  equal  to  the 
area  of  the  spherical  segment.     In  other 


Tij.30. 


words,  the  zone's  surface  is  ecjual  to  the  zpne'.  altitude 


(?  cyliiuler  also  be 
'es."  "  And  with 
lur  first  supposi- 
1  length  in  both 
)  of  that  cylinder 
e  sphere,  for  both 
38  of  light."     l^ot 


must."  [V\g.  •:i9.] 
11  this  in  mind  it 
derstand  how  the 
it  is,  can  be  repre- 
charts. 

an  important  point 
with  that  lesson — 
J  uniformly  illumi- 
light  streams  forth 
1  its  centre.    Thus 
;  a  familiar  method 
a  spherical  zone  or 
ing  a  sphere  within 
and  producing  the 
)lanes  which  bound 
liey  cut  off  a  section 
!  area  of  that  cylin- 
viously  equal  to  the 
segment.     In  other 
the  zQne'3  altitude 


A  CLASS  IN   GEOMETRY. 


43 


multiplied  by  the  circumference  of  a  great  circle  of  its 
^^'^e^eL'o^VinitetoIndependentObservation.-^^^^^ 
a  voar  had  now  elapsed  since  the  formation  of  oui  little 
class   a       our  progress  was  gratifying.    The  eldest  boy 
ri)  "un  t  0  Lu  of  Euclid  at  school,  and  was  earning 
;  common  marks  for  his  proticioncy.     In  the  lesson    I 
n  led,  and  in  others  which  followed  them  al    tlie 
ds.     wed  their  interest  by  being  constantly  on  the  loi^. 
out  for  new  illustrations.     Let  an  instance  or  two  of  this 
8  mce      0..  day  they  walked  to  an  immense  sugar-re- 
fit^  some  distance  off,  paced  around  it,  -t.niated  it 
hei.^ht  and  brought  me  their  calculations  as  to  its  storage 
ca  :^   y  in  comparison  with  that  of  a  small  warehouse 
2"ty,-calculations  showing  how  much  outer  wa     and 
roo    wL  saved  in  the  vast  proportions  of  the  rehnery. 
At  hie  an  extension  of  the  house  was  heated  in  wmter 
by  a  small  stove;  at  a  neighboring  station  of  the  s  reet- 
rlilvay  there  wa;  a  much  larger  stove  of  the  same  pat  term 
Countin.^  etliciency  to  depend  on  surface,  one  of  the  boys 
asked  me  if  it  would  not  be  better  to  have  two  small  stove 
iuste-vd  of  the  large  one.     He  was  perfectly  conversan 
"iUt^ie  reason  why  steam-fitters  use  small  pipes  for  their 
bating  coils,  and  why  their  radiators  abound  "^  l^-^«  -^ 
idl.    He   aw,  for  himself,  why  in  buying  a  tm  bak  ng- 
0 1  one  of  cubical  form  had  been  chosen-he  knew  t^iat 
'  ,  oven  of  that  shape  would  throw  out  waste  uUy  and 
h-  vn  Mly  less  heat  thin  if  it  were  made  in  any  other  orm 
0       no  ning  our  neighbor's  conservatory  had  more  than  a 
tZnot  its  panes  of  glass  broken  by  Ija.ls  ones.     My 
elde     pnpil  came  to  me  with  the  reason  why  the  had  had 
been  so  destructive.    Certain  of  its  pellets  had  cohered  ^ 
form  masses  of  ice  nearly  half  an  inch  in  diameter.     These 
mlirTess  resisted  by  the  air,  and  heavier  than  separate 
pelS  had  fallen  with  the  force  of  bullets.    One  summer 


44 


A  CLASS  IN   CEOMRTRY. 


(liiy  the  boys  and  I  ciuno  to  n  liay-ticld ;  part  of  the  hay  was 
beiiis  nuulo,  ami  part  had  bet-n  built  into  stacks.  They 
pointed  out  that  tlio  farmers  in  nuikinj,'  hay  tedded  it  out 
to  get  uU  the  light  and  air  it  loulu,  and  that  when  the  liay 
was  thoroughly  dry  it  was  kept  from  rain  and  frost  in  com- 
pactly built  stacks  of  huuiII  exposed  surface. 

Laws  are  Strings  wherewith  to  tie  Facts  together.— 
It  may  be  no  more  than  the  elTect  of  bias  due  to  an  indi- 
vidual preference  for  the  study,  but,  in  the  light  of  its 
iniluence  on  these  three  young  minds,  I  cannot  help  think- 
ing that  geometry  alTords  a  happy  means  of  developing 
and  directing  the  powers  of  observation  and  reasoning. 
Our  lessons,  simple  as  they  were,  had  shown  us  that  a 
single  elementary  principle  such  as  that  subsisting  between 
"arfaces  and  volumes  may  bind  together  a  wide  variety  of 
fact  in  physics,  chemistry,  astronomy,  navigation,  and  en- 
gineering.    Young  people  are  ajjt  to  imagine  that  a  most 
formidable  barrier    rises   between    science    and   common 
things:    we  had  discovered  that  no  such  barrier  exists. 
When  the  boys  came  to  study  minerals,  plants,  and  insects, 
they  found  that  they  could  gainfully  adopt  the  method 
with  which  they  had  taken  up  geometry— their  collections 
had  value,  and  were  added  to  with  ever-renewed  interest, 
because  they  kept  in  mind  a  connecting  thread  of  classifi- 
cation, instead  of  accumulating  at  random  a  variety  of 
merely  curious  things.     One  of  the  boys  gathered  speci- 
mens of  materials  used  for  building  in  the  adjoining  city, 
and  could  show  limestones  and  sandstones  as  they  leave 
the  quarry  and  are  prepared  by  the  mason  with  hammer 
and  chisel.    His  collection  included  bricks  burned  and 
uuburned,  mouldings  in  terra  cotta,  ornamental  glazed 
tiles,  and  specimens  of  the  rarer  cabinet  woods. 

Invention William    George    Spencer,  the   father   of 

Herbert  Spencer,  in  a  little  book  entitled  "Inventional 


A  CLASS  IN   GEOMETRY. 


45 


part  of  the  hay  wus 
into  stacks.  They 
r  hay  tedded  it  out 

that  when  the  liay 
n  and  frost  iu  com- 
fiico. 

Facts  together.— 
lius  duo  to  an  indi- 
in  the  light  of  its 

cannot  help  think- 
eans  of  developing 
ion  and  reasoning. 
I  shown  us  that  a 

subsisting  between 
er  a  wide  variety  of 
navigation,  and  en« 
imagine  that  a  most 
ence  and  common 
iuch  barrier  exists. 
,  plants,  and  insects, 

adopt  the  method 
py — their  collections 
er-renewed  interest, 
s  thread  of  classifi- 
mdom  a  variety  of 
)oys  gathered  speci- 

tlie  adjoining  city, 
itoncs  as  they  leave 
nason  with  hammer 

bricks  burned  and 

ornamental  glazed 
et  woods. 

icer,  the   father   of 
ititled  "Inveutional 


Geometry"  h^s  shown  how  geometry  can  be  taught  so  as 
to  educe  the  noble  faculty  of  invention.     At  the   Ihgl. 
School  in  Yonkers.  New  York,  I  have  see.i  or.g.na    and 
most     beautiful    solutions    of     Mr.    Spencers     problems 
worked  out  by  the  pupils.      I  d,.ubt  not    that    n.  other 
fields  of  study  a  lively  interest  can  be  aroused  by  keeping 
before  learners  an  important  princi..lo  of  nati.rc  or  ar - 
and  inciting  to  the  sean^h  for  illustrations  of  it.     Su<;h  a 
principle  may  serve  as  a  (inder-thougbt  in  cleanng  away  a 
ditticulty  not  otherwise  to  be  surmounted.     Originality, 
ingenuity,  can  thus  be  drawn  out,  ad.ling  to  the  resource- 
fulness  of  a  student's   mind.     At   the    High   School   m 
Yonkers  and  at  other  excellent  schools  the  literary  exer- 
cises make  an  appeal  to  inventiveness,  and  not  less  elTec- 
tively  than  the   lessons  in  geometry.      A  teiicher  tells  a 
story  to  the  extent  of  one  half;  the  remauuler  has  to  bo 
written   from    the    imagination   of    the   l'«^;'«'er«.      Ihis 
n.ethod   is   fully   illustrated    in    Professor   h    U.   Shaw  s 
recently  published  "  English   Composition   by  Practice. 
In  several  newly  established  schools  of  industrial  art  there 
is  a  somewhat  similar  challenge  to  ingenuity.     A  teacher 
draws  a  few  straight  lines  and  simple  curves  upon  the 
blacl  board,  and  after  a  minute  or  two  rubs  them  out.     His 
pupils  then  repeat  them  on  their  slates  from  memory,  and 
combine  them  to  form  original  designs.     A  1  this  is  good 
It  makes  for  the  development  of  individual  aptitude.     It 
gives  a  punil  a  pleasing  sense  of  having,  in  however  small 
a  way,  added  to  knowledge  or  to  art;  while,  incidentally, 
the  knowledge  or  the  art  is  the  better  grasped  and  ce- 
mented in  the  act  of  making  this  addition,  inconsiderable 
though  it  be.     Besides  the  direct  gain  which  comes  of 
brin  Ang  out  the  ingenuity  of  boys  and  girls  in  these  ways 
and  In  all  ways,  there  is  an  indirect  gain  not  less  impor- 
tant-in  laying  the  foundation  for  an  intelligent  sympathy 


4« 


A  CLASS  IN   GEOMUTRY. 


witli  the  labors  of  tho  inventor,  the  discoveror,  tl.c  man  of 
originality  who  finds  a  now  and  good  way  ont  of  a  ditli- 
culty  indnstrial,  social,  or  political.  Only  when  all  tho 
people  are  informed  as  to  the  gifts  these  men  stand  ready 
to  confer,  only  when  they  are  heartily  willing  to  receive 
those  gifts,  cau  civilization  come  to  its  best  estate. 


veror,  tho  man  of 
iiy  out  of  a  (lilli- 
nly  when  all  tlio 
men  stand  ready 
villhig  to  receive 
38t  estate. 


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riodicals. 


it  In  its  33rd  vear. 

irculated  eilucatlonal 
;j  with  iJeai  thai  will 
education.  The  best 
uhing  is  found  in  ll 
ssorcd  «)Ut  of  other 
wn  special  writers— 


fune  at  |i.oo  a  yeac 
being  devoted  almost 
ind  devices.  Several 
ilue. 

1TUTE 

t  is  edited  intht  same 
HE  Journal,  and  has 
ic  most  populcfr  educa- 
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taught  by  the  teacher 

IONS. 

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te  Examination  Quea* 


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tndftr  a  fru  tamfU. 


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L_. 


